TPTP Problem File: ITP139^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP139^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Paraconsistency problem prob_881__3274616_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Paraconsistency/prob_881__3274616_1 [Des21]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 489 ( 97 unt; 133 typ; 0 def)
% Number of atoms : 1235 ( 485 equ; 0 cnn)
% Maximal formula atoms : 9 ( 3 avg)
% Number of connectives : 3051 ( 29 ~; 1 |; 47 &;2368 @)
% ( 0 <=>; 606 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 679 ( 679 >; 0 *; 0 +; 0 <<)
% Number of symbols : 120 ( 119 usr; 7 con; 0-3 aty)
% Number of variables : 1095 ( 49 ^;1006 !; 40 ?;1095 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:39:29.911
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
set_Pr2014382200lle_tv: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
produc1647334210lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_J,type,
set_Pr1725653840tv_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
set_Pr13445366lle_tv: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
produc194167194tv_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
produc270694976lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
set_se800614875lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1986765409at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
set_Pa1764573435lle_tv: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
paraco1605129243lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (119)
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
inj_on1955349912lle_tv: ( nat > paraco1605129243lle_tv ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat,type,
inj_on1179712058tv_nat: ( paraco1605129243lle_tv > nat ) > set_Pa1764573435lle_tv > $o ).
thf(sy_c_Fun_Oinj__on_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
inj_on772319074lle_tv: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > set_Pa1764573435lle_tv > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
inj_on642995232at_nat: ( product_prod_nat_nat > nat ) > set_Pr1986765409at_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
inj_on520021963lle_tv: ( product_prod_nat_nat > paraco1605129243lle_tv ) > set_Pr1986765409at_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on26176689at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1986765409at_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
inj_on1840679568lle_tv: ( product_prod_nat_nat > produc270694976lle_tv ) > set_Pr1986765409at_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
inj_on1764151786tv_nat: ( product_prod_nat_nat > produc194167194tv_nat ) > set_Pr1986765409at_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Nat__Onat,type,
inj_on936843139tv_nat: ( produc270694976lle_tv > nat ) > set_Pr13445366lle_tv > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
inj_on394919723lle_tv: ( produc270694976lle_tv > paraco1605129243lle_tv ) > set_Pr13445366lle_tv > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on1954136462at_nat: ( produc270694976lle_tv > product_prod_nat_nat ) > set_Pr13445366lle_tv > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
inj_on1745913142lle_tv: ( produc270694976lle_tv > produc270694976lle_tv ) > set_Pr13445366lle_tv > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Nat__Onat,type,
inj_on808253149at_nat: ( produc194167194tv_nat > nat ) > set_Pr1725653840tv_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
inj_on1612474245lle_tv: ( produc194167194tv_nat > paraco1605129243lle_tv ) > set_Pr1725653840tv_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on1971554740at_nat: ( produc194167194tv_nat > product_prod_nat_nat ) > set_Pr1725653840tv_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
inj_on552895452tv_nat: ( produc194167194tv_nat > produc194167194tv_nat ) > set_Pr1725653840tv_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
inj_on967082628lle_tv: ( produc194167194tv_nat > produc1647334210lle_tv ) > set_Pr1725653840tv_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
inj_on2112487900lle_tv: ( produc1647334210lle_tv > produc1647334210lle_tv ) > set_Pr2014382200lle_tv > $o ).
thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
strict2146030304at_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
strict1310998166et_nat: ( nat > set_nat ) > set_nat > $o ).
thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
strict1658340843lle_tv: ( nat > set_Pa1764573435lle_tv ) > set_nat > $o ).
thf(sy_c_Fun_Ostrict__mono__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
strict1802283670at_nat: ( set_nat > nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Ostrict__mono__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
strict473775692et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Ostrict__mono__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
strict172559285lle_tv: ( set_nat > set_Pa1764573435lle_tv ) > set_set_nat > $o ).
thf(sy_c_Fun_Ostrict__mono__on_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Nat__Onat,type,
strict1125660301tv_nat: ( set_Pa1764573435lle_tv > nat ) > set_se800614875lle_tv > $o ).
thf(sy_c_Fun_Ostrict__mono__on_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Set__Oset_It__Nat__Onat_J,type,
strict1455779395et_nat: ( set_Pa1764573435lle_tv > set_nat ) > set_se800614875lle_tv > $o ).
thf(sy_c_Fun_Ostrict__mono__on_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
strict592849515lle_tv: ( set_Pa1764573435lle_tv > set_Pa1764573435lle_tv ) > set_se800614875lle_tv > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
the_in138087638lle_tv: set_nat > ( nat > paraco1605129243lle_tv ) > paraco1605129243lle_tv > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat,type,
the_in1509933432tv_nat: set_Pa1764573435lle_tv > ( paraco1605129243lle_tv > nat ) > nat > paraco1605129243lle_tv ).
thf(sy_c_Fun_Othe__inv__into_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
the_in452116896lle_tv: set_Pa1764573435lle_tv > ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > paraco1605129243lle_tv > paraco1605129243lle_tv ).
thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
the_in2068894451at_nat: set_Pr1986765409at_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
the_in744013006lle_tv: set_Pr1986765409at_nat > ( product_prod_nat_nat > produc270694976lle_tv ) > produc270694976lle_tv > product_prod_nat_nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
the_in667485224tv_nat: set_Pr1986765409at_nat > ( product_prod_nat_nat > produc194167194tv_nat ) > produc194167194tv_nat > product_prod_nat_nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
the_in857469900at_nat: set_Pr13445366lle_tv > ( produc270694976lle_tv > product_prod_nat_nat ) > product_prod_nat_nat > produc270694976lle_tv ).
thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
the_in1388677748lle_tv: set_Pr13445366lle_tv > ( produc270694976lle_tv > produc270694976lle_tv ) > produc270694976lle_tv > produc270694976lle_tv ).
thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
the_in874888178at_nat: set_Pr1725653840tv_nat > ( produc194167194tv_nat > product_prod_nat_nat ) > product_prod_nat_nat > produc194167194tv_nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
the_in195660058tv_nat: set_Pr1725653840tv_nat > ( produc194167194tv_nat > produc194167194tv_nat ) > produc194167194tv_nat > produc194167194tv_nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
the_in427240730lle_tv: set_Pr2014382200lle_tv > ( produc1647334210lle_tv > produc1647334210lle_tv ) > produc1647334210lle_tv > produc1647334210lle_tv ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
ord_le727057319lle_tv: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
ord_le241064091lle_tv: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
order_769474267at_nat: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Paraconsistency____Mirabelle____qxuposogha__Otv_M_Eo_J,type,
top_to1299250770e_tv_o: paraco1605129243lle_tv > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
top_to632227436_nat_o: product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_M_Eo_J,type,
top_to1889334367e_tv_o: produc270694976lle_tv > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_M_Eo_J,type,
top_to900819589_nat_o: produc194167194tv_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
top_to1057771083lle_tv: set_Pa1764573435lle_tv ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to1555900337at_nat: set_Pr1986765409at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
top_to1411004326lle_tv: set_Pr13445366lle_tv ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_J,type,
top_to975729152tv_nat: set_Pr1725653840tv_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
top_to1871154984lle_tv: set_Pr2014382200lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ochange__tv,type,
paraco1147068288nge_tv: ( nat > nat ) > paraco1605129243lle_tv > paraco1605129243lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Otv_OIndet,type,
paraco793048302_Indet: nat > paraco1605129243lle_tv ).
thf(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
produc1320959600at_nat: ( nat > nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
produc237097339lle_tv: ( nat > nat ) > produc194167194tv_nat > produc194167194tv_nat ).
thf(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat,type,
produc1608943133tv_nat: ( nat > paraco1605129243lle_tv ) > product_prod_nat_nat > produc270694976lle_tv ).
thf(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
produc1804670661lle_tv: ( nat > paraco1605129243lle_tv ) > produc194167194tv_nat > produc1647334210lle_tv ).
thf(sy_c_Product__Type_Oapsnd_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat,type,
produc2040316391tv_nat: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > produc270694976lle_tv > produc270694976lle_tv ).
thf(sy_c_Product__Type_Oapsnd_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
produc1408152463lle_tv: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > produc1647334210lle_tv > produc1647334210lle_tv ).
thf(sy_c_Product__Type_Omap__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
produc505105722at_nat: ( nat > nat ) > ( nat > nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Omap__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
produc2101297531lle_tv: ( nat > nat ) > ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > produc270694976lle_tv > produc270694976lle_tv ).
thf(sy_c_Product__Type_Omap__prod_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat,type,
produc189459613tv_nat: ( nat > paraco1605129243lle_tv ) > ( paraco1605129243lle_tv > nat ) > produc270694976lle_tv > produc194167194tv_nat ).
thf(sy_c_Product__Type_Omap__prod_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
produc1975500101lle_tv: ( nat > paraco1605129243lle_tv ) > ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > produc270694976lle_tv > produc1647334210lle_tv ).
thf(sy_c_Product__Type_Omap__prod_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
produc386540311at_nat: ( paraco1605129243lle_tv > nat ) > ( nat > nat ) > produc194167194tv_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Omap__prod_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
produc72207551lle_tv: ( paraco1605129243lle_tv > nat ) > ( nat > paraco1605129243lle_tv ) > produc194167194tv_nat > produc270694976lle_tv ).
thf(sy_c_Product__Type_Omap__prod_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat,type,
produc51277375tv_nat: ( paraco1605129243lle_tv > nat ) > ( paraco1605129243lle_tv > nat ) > produc1647334210lle_tv > product_prod_nat_nat ).
thf(sy_c_Product__Type_Omap__prod_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
produc1095080935lle_tv: ( paraco1605129243lle_tv > nat ) > ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > produc1647334210lle_tv > produc270694976lle_tv ).
thf(sy_c_Product__Type_Omap__prod_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat_001t__Nat__Onat,type,
produc42536895at_nat: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > ( nat > nat ) > produc194167194tv_nat > produc194167194tv_nat ).
thf(sy_c_Product__Type_Omap__prod_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
produc1521089895lle_tv: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > ( nat > paraco1605129243lle_tv ) > produc194167194tv_nat > produc1647334210lle_tv ).
thf(sy_c_Product__Type_Omap__prod_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat,type,
produc1500159719tv_nat: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > ( paraco1605129243lle_tv > nat ) > produc1647334210lle_tv > produc194167194tv_nat ).
thf(sy_c_Product__Type_Omap__prod_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
produc1345917071lle_tv: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > produc1647334210lle_tv > produc1647334210lle_tv ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
collec1111687558lle_tv: ( paraco1605129243lle_tv > $o ) > set_Pa1764573435lle_tv ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
collec924990229lle_tv: ( produc270694976lle_tv > $o ) > set_Pr13445366lle_tv ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
collec848462447tv_nat: ( produc194167194tv_nat > $o ) > set_Pr1725653840tv_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
image_1151963140lle_tv: ( nat > paraco1605129243lle_tv ) > set_nat > set_Pa1764573435lle_tv ).
thf(sy_c_Set_Oimage_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat,type,
image_376325286tv_nat: ( paraco1605129243lle_tv > nat ) > set_Pa1764573435lle_tv > set_nat ).
thf(sy_c_Set_Oimage_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
image_335818702lle_tv: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > set_Pa1764573435lle_tv > set_Pa1764573435lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_1397137548at_nat: ( product_prod_nat_nat > nat ) > set_Pr1986765409at_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
image_1093035615lle_tv: ( product_prod_nat_nat > paraco1605129243lle_tv ) > set_Pr1986765409at_nat > set_Pa1764573435lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_815163717at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1986765409at_nat > set_Pr1986765409at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_1674248956lle_tv: ( product_prod_nat_nat > produc270694976lle_tv ) > set_Pr1986765409at_nat > set_Pr13445366lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
image_1597721174tv_nat: ( product_prod_nat_nat > produc194167194tv_nat ) > set_Pr1986765409at_nat > set_Pr1725653840tv_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Nat__Onat,type,
image_1509856791tv_nat: ( produc270694976lle_tv > nat ) > set_Pr13445366lle_tv > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
image_802013119lle_tv: ( produc270694976lle_tv > paraco1605129243lle_tv ) > set_Pr13445366lle_tv > set_Pa1764573435lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1787705850at_nat: ( produc270694976lle_tv > product_prod_nat_nat ) > set_Pr13445366lle_tv > set_Pr1986765409at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_149830562lle_tv: ( produc270694976lle_tv > produc270694976lle_tv ) > set_Pr13445366lle_tv > set_Pr13445366lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
image_53754658tv_nat: ( produc270694976lle_tv > produc194167194tv_nat ) > set_Pr13445366lle_tv > set_Pr1725653840tv_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_988201162lle_tv: ( produc270694976lle_tv > produc1647334210lle_tv ) > set_Pr13445366lle_tv > set_Pr2014382200lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_1381266801at_nat: ( produc194167194tv_nat > nat ) > set_Pr1725653840tv_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
image_2019567641lle_tv: ( produc194167194tv_nat > paraco1605129243lle_tv ) > set_Pr1725653840tv_nat > set_Pa1764573435lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1805124128at_nat: ( produc194167194tv_nat > product_prod_nat_nat ) > set_Pr1725653840tv_nat > set_Pr1986765409at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_1200372424lle_tv: ( produc194167194tv_nat > produc270694976lle_tv ) > set_Pr1725653840tv_nat > set_Pr13445366lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
image_1104296520tv_nat: ( produc194167194tv_nat > produc194167194tv_nat ) > set_Pr1725653840tv_nat > set_Pr1725653840tv_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_228488944lle_tv: ( produc194167194tv_nat > produc1647334210lle_tv ) > set_Pr1725653840tv_nat > set_Pr2014382200lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_901203832at_nat: ( produc1647334210lle_tv > product_prod_nat_nat ) > set_Pr2014382200lle_tv > set_Pr1986765409at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_1892891680lle_tv: ( produc1647334210lle_tv > produc270694976lle_tv ) > set_Pr2014382200lle_tv > set_Pr13445366lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
image_1796815776tv_nat: ( produc1647334210lle_tv > produc194167194tv_nat ) > set_Pr2014382200lle_tv > set_Pr1725653840tv_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_1221801032lle_tv: ( produc1647334210lle_tv > produc1647334210lle_tv ) > set_Pr2014382200lle_tv > set_Pr2014382200lle_tv ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
member266900804lle_tv: paraco1605129243lle_tv > set_Pa1764573435lle_tv > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
member813679063lle_tv: produc270694976lle_tv > set_Pr13445366lle_tv > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
member737151281tv_nat: produc194167194tv_nat > set_Pr1725653840tv_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
member1531318361lle_tv: produc1647334210lle_tv > set_Pr2014382200lle_tv > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
member1875402276lle_tv: set_Pa1764573435lle_tv > set_se800614875lle_tv > $o ).
thf(sy_v_f,type,
f: nat > nat ).
% Relevant facts (355)
thf(fact_0__092_060open_062_092_060And_062tv2_Atv1_O_Achange__tv_Af_Atv1_A_061_Achange__tv_Af_Atv2_A_092_060Longrightarrow_062_Atv1_A_061_Atv2_092_060close_062,axiom,
! [Tv1: paraco1605129243lle_tv,Tv2: paraco1605129243lle_tv] :
( ( ( paraco1147068288nge_tv @ f @ Tv1 )
= ( paraco1147068288nge_tv @ f @ Tv2 ) )
=> ( Tv1 = Tv2 ) ) ).
% \<open>\<And>tv2 tv1. change_tv f tv1 = change_tv f tv2 \<Longrightarrow> tv1 = tv2\<close>
thf(fact_1_UNIV__I,axiom,
! [X: produc194167194tv_nat] : ( member737151281tv_nat @ X @ top_to975729152tv_nat ) ).
% UNIV_I
thf(fact_2_UNIV__I,axiom,
! [X: produc270694976lle_tv] : ( member813679063lle_tv @ X @ top_to1411004326lle_tv ) ).
% UNIV_I
thf(fact_3_UNIV__I,axiom,
! [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ top_to1555900337at_nat ) ).
% UNIV_I
thf(fact_4_UNIV__I,axiom,
! [X: paraco1605129243lle_tv] : ( member266900804lle_tv @ X @ top_to1057771083lle_tv ) ).
% UNIV_I
thf(fact_5_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_6_iso__tuple__UNIV__I,axiom,
! [X: produc194167194tv_nat] : ( member737151281tv_nat @ X @ top_to975729152tv_nat ) ).
% iso_tuple_UNIV_I
thf(fact_7_iso__tuple__UNIV__I,axiom,
! [X: produc270694976lle_tv] : ( member813679063lle_tv @ X @ top_to1411004326lle_tv ) ).
% iso_tuple_UNIV_I
thf(fact_8_iso__tuple__UNIV__I,axiom,
! [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ top_to1555900337at_nat ) ).
% iso_tuple_UNIV_I
thf(fact_9_iso__tuple__UNIV__I,axiom,
! [X: paraco1605129243lle_tv] : ( member266900804lle_tv @ X @ top_to1057771083lle_tv ) ).
% iso_tuple_UNIV_I
thf(fact_10_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_11_top__apply,axiom,
( top_to1299250770e_tv_o
= ( ^ [X2: paraco1605129243lle_tv] : top_top_o ) ) ).
% top_apply
thf(fact_12_top__apply,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : top_top_o ) ) ).
% top_apply
thf(fact_13_injD,axiom,
! [F: produc1647334210lle_tv > produc1647334210lle_tv,X: produc1647334210lle_tv,Y: produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F @ top_to1871154984lle_tv )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_14_injD,axiom,
! [F: nat > paraco1605129243lle_tv,X: nat,Y: nat] :
( ( inj_on1955349912lle_tv @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_15_injD,axiom,
! [F: produc194167194tv_nat > produc194167194tv_nat,X: produc194167194tv_nat,Y: produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F @ top_to975729152tv_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_16_injD,axiom,
! [F: produc270694976lle_tv > produc270694976lle_tv,X: produc270694976lle_tv,Y: produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F @ top_to1411004326lle_tv )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_17_injD,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F @ top_to1555900337at_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_18_injD,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F @ top_to1057771083lle_tv )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_19_injD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_20_injI,axiom,
! [F: produc1647334210lle_tv > produc1647334210lle_tv] :
( ! [X3: produc1647334210lle_tv,Y2: produc1647334210lle_tv] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) )
=> ( inj_on2112487900lle_tv @ F @ top_to1871154984lle_tv ) ) ).
% injI
thf(fact_21_injI,axiom,
! [F: nat > paraco1605129243lle_tv] :
( ! [X3: nat,Y2: nat] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) )
=> ( inj_on1955349912lle_tv @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_22_injI,axiom,
! [F: produc194167194tv_nat > produc194167194tv_nat] :
( ! [X3: produc194167194tv_nat,Y2: produc194167194tv_nat] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) )
=> ( inj_on552895452tv_nat @ F @ top_to975729152tv_nat ) ) ).
% injI
thf(fact_23_injI,axiom,
! [F: produc270694976lle_tv > produc270694976lle_tv] :
( ! [X3: produc270694976lle_tv,Y2: produc270694976lle_tv] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) )
=> ( inj_on1745913142lle_tv @ F @ top_to1411004326lle_tv ) ) ).
% injI
thf(fact_24_injI,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat] :
( ! [X3: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) )
=> ( inj_on26176689at_nat @ F @ top_to1555900337at_nat ) ) ).
% injI
thf(fact_25_injI,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [X3: paraco1605129243lle_tv,Y2: paraco1605129243lle_tv] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) )
=> ( inj_on772319074lle_tv @ F @ top_to1057771083lle_tv ) ) ).
% injI
thf(fact_26_injI,axiom,
! [F: nat > nat] :
( ! [X3: nat,Y2: nat] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_27_inj__eq,axiom,
! [F: produc1647334210lle_tv > produc1647334210lle_tv,X: produc1647334210lle_tv,Y: produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F @ top_to1871154984lle_tv )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_28_inj__eq,axiom,
! [F: nat > paraco1605129243lle_tv,X: nat,Y: nat] :
( ( inj_on1955349912lle_tv @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_29_inj__eq,axiom,
! [F: produc194167194tv_nat > produc194167194tv_nat,X: produc194167194tv_nat,Y: produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F @ top_to975729152tv_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_30_inj__eq,axiom,
! [F: produc270694976lle_tv > produc270694976lle_tv,X: produc270694976lle_tv,Y: produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F @ top_to1411004326lle_tv )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_31_inj__eq,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F @ top_to1555900337at_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_32_inj__eq,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F @ top_to1057771083lle_tv )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_33_inj__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_34_inj__def,axiom,
! [F: produc1647334210lle_tv > produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F @ top_to1871154984lle_tv )
= ( ! [X2: produc1647334210lle_tv,Y3: produc1647334210lle_tv] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ).
% inj_def
thf(fact_35_inj__def,axiom,
! [F: nat > paraco1605129243lle_tv] :
( ( inj_on1955349912lle_tv @ F @ top_top_set_nat )
= ( ! [X2: nat,Y3: nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ).
% inj_def
thf(fact_36_inj__def,axiom,
! [F: produc194167194tv_nat > produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F @ top_to975729152tv_nat )
= ( ! [X2: produc194167194tv_nat,Y3: produc194167194tv_nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ).
% inj_def
thf(fact_37_inj__def,axiom,
! [F: produc270694976lle_tv > produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F @ top_to1411004326lle_tv )
= ( ! [X2: produc270694976lle_tv,Y3: produc270694976lle_tv] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ).
% inj_def
thf(fact_38_inj__def,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F @ top_to1555900337at_nat )
= ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ).
% inj_def
thf(fact_39_inj__def,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F @ top_to1057771083lle_tv )
= ( ! [X2: paraco1605129243lle_tv,Y3: paraco1605129243lle_tv] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ).
% inj_def
thf(fact_40_inj__def,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
= ( ! [X2: nat,Y3: nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ).
% inj_def
thf(fact_41_inj__onD,axiom,
! [F: produc1647334210lle_tv > produc1647334210lle_tv,A: set_Pr2014382200lle_tv,X: produc1647334210lle_tv,Y: produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member1531318361lle_tv @ X @ A )
=> ( ( member1531318361lle_tv @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_42_inj__onD,axiom,
! [F: produc194167194tv_nat > produc194167194tv_nat,A: set_Pr1725653840tv_nat,X: produc194167194tv_nat,Y: produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member737151281tv_nat @ X @ A )
=> ( ( member737151281tv_nat @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_43_inj__onD,axiom,
! [F: produc270694976lle_tv > produc270694976lle_tv,A: set_Pr13445366lle_tv,X: produc270694976lle_tv,Y: produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member813679063lle_tv @ X @ A )
=> ( ( member813679063lle_tv @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_44_inj__onD,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A: set_Pr1986765409at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member701585322at_nat @ X @ A )
=> ( ( member701585322at_nat @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_45_inj__onD,axiom,
! [F: nat > paraco1605129243lle_tv,A: set_nat,X: nat,Y: nat] :
( ( inj_on1955349912lle_tv @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_46_inj__onD,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member266900804lle_tv @ X @ A )
=> ( ( member266900804lle_tv @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_47_inj__onD,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_48_inj__onI,axiom,
! [A: set_Pr2014382200lle_tv,F: produc1647334210lle_tv > produc1647334210lle_tv] :
( ! [X3: produc1647334210lle_tv,Y2: produc1647334210lle_tv] :
( ( member1531318361lle_tv @ X3 @ A )
=> ( ( member1531318361lle_tv @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) )
=> ( inj_on2112487900lle_tv @ F @ A ) ) ).
% inj_onI
thf(fact_49_inj__onI,axiom,
! [A: set_Pr1725653840tv_nat,F: produc194167194tv_nat > produc194167194tv_nat] :
( ! [X3: produc194167194tv_nat,Y2: produc194167194tv_nat] :
( ( member737151281tv_nat @ X3 @ A )
=> ( ( member737151281tv_nat @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) )
=> ( inj_on552895452tv_nat @ F @ A ) ) ).
% inj_onI
thf(fact_50_inj__onI,axiom,
! [A: set_Pr13445366lle_tv,F: produc270694976lle_tv > produc270694976lle_tv] :
( ! [X3: produc270694976lle_tv,Y2: produc270694976lle_tv] :
( ( member813679063lle_tv @ X3 @ A )
=> ( ( member813679063lle_tv @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) )
=> ( inj_on1745913142lle_tv @ F @ A ) ) ).
% inj_onI
thf(fact_51_inj__onI,axiom,
! [A: set_Pr1986765409at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
( ! [X3: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( member701585322at_nat @ X3 @ A )
=> ( ( member701585322at_nat @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) )
=> ( inj_on26176689at_nat @ F @ A ) ) ).
% inj_onI
thf(fact_52_inj__onI,axiom,
! [A: set_nat,F: nat > paraco1605129243lle_tv] :
( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) )
=> ( inj_on1955349912lle_tv @ F @ A ) ) ).
% inj_onI
thf(fact_53_inj__onI,axiom,
! [A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [X3: paraco1605129243lle_tv,Y2: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ A )
=> ( ( member266900804lle_tv @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) )
=> ( inj_on772319074lle_tv @ F @ A ) ) ).
% inj_onI
thf(fact_54_inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_onI
thf(fact_55_inj__on__def,axiom,
( inj_on2112487900lle_tv
= ( ^ [F2: produc1647334210lle_tv > produc1647334210lle_tv,A2: set_Pr2014382200lle_tv] :
! [X2: produc1647334210lle_tv] :
( ( member1531318361lle_tv @ X2 @ A2 )
=> ! [Y3: produc1647334210lle_tv] :
( ( member1531318361lle_tv @ Y3 @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_56_inj__on__def,axiom,
( inj_on552895452tv_nat
= ( ^ [F2: produc194167194tv_nat > produc194167194tv_nat,A2: set_Pr1725653840tv_nat] :
! [X2: produc194167194tv_nat] :
( ( member737151281tv_nat @ X2 @ A2 )
=> ! [Y3: produc194167194tv_nat] :
( ( member737151281tv_nat @ Y3 @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_57_inj__on__def,axiom,
( inj_on1745913142lle_tv
= ( ^ [F2: produc270694976lle_tv > produc270694976lle_tv,A2: set_Pr13445366lle_tv] :
! [X2: produc270694976lle_tv] :
( ( member813679063lle_tv @ X2 @ A2 )
=> ! [Y3: produc270694976lle_tv] :
( ( member813679063lle_tv @ Y3 @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_58_inj__on__def,axiom,
( inj_on26176689at_nat
= ( ^ [F2: product_prod_nat_nat > product_prod_nat_nat,A2: set_Pr1986765409at_nat] :
! [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ A2 )
=> ! [Y3: product_prod_nat_nat] :
( ( member701585322at_nat @ Y3 @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_59_inj__on__def,axiom,
( inj_on1955349912lle_tv
= ( ^ [F2: nat > paraco1605129243lle_tv,A2: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_60_inj__on__def,axiom,
( inj_on772319074lle_tv
= ( ^ [F2: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv] :
! [X2: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X2 @ A2 )
=> ! [Y3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ Y3 @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_61_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A2: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_62_inj__on__cong,axiom,
! [A: set_Pr2014382200lle_tv,F: produc1647334210lle_tv > produc1647334210lle_tv,G: produc1647334210lle_tv > produc1647334210lle_tv] :
( ! [A3: produc1647334210lle_tv] :
( ( member1531318361lle_tv @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on2112487900lle_tv @ F @ A )
= ( inj_on2112487900lle_tv @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_63_inj__on__cong,axiom,
! [A: set_Pr1725653840tv_nat,F: produc194167194tv_nat > produc194167194tv_nat,G: produc194167194tv_nat > produc194167194tv_nat] :
( ! [A3: produc194167194tv_nat] :
( ( member737151281tv_nat @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on552895452tv_nat @ F @ A )
= ( inj_on552895452tv_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_64_inj__on__cong,axiom,
! [A: set_Pr13445366lle_tv,F: produc270694976lle_tv > produc270694976lle_tv,G: produc270694976lle_tv > produc270694976lle_tv] :
( ! [A3: produc270694976lle_tv] :
( ( member813679063lle_tv @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on1745913142lle_tv @ F @ A )
= ( inj_on1745913142lle_tv @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_65_inj__on__cong,axiom,
! [A: set_Pr1986765409at_nat,F: product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat] :
( ! [A3: product_prod_nat_nat] :
( ( member701585322at_nat @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on26176689at_nat @ F @ A )
= ( inj_on26176689at_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_66_inj__on__cong,axiom,
! [A: set_nat,F: nat > paraco1605129243lle_tv,G: nat > paraco1605129243lle_tv] :
( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on1955349912lle_tv @ F @ A )
= ( inj_on1955349912lle_tv @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_67_inj__on__cong,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
= ( inj_on_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_68_inj__on__cong,axiom,
! [A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv,G: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [A3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on772319074lle_tv @ F @ A )
= ( inj_on772319074lle_tv @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_69_inj__on__eq__iff,axiom,
! [F: produc1647334210lle_tv > produc1647334210lle_tv,A: set_Pr2014382200lle_tv,X: produc1647334210lle_tv,Y: produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F @ A )
=> ( ( member1531318361lle_tv @ X @ A )
=> ( ( member1531318361lle_tv @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_70_inj__on__eq__iff,axiom,
! [F: produc194167194tv_nat > produc194167194tv_nat,A: set_Pr1725653840tv_nat,X: produc194167194tv_nat,Y: produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F @ A )
=> ( ( member737151281tv_nat @ X @ A )
=> ( ( member737151281tv_nat @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_71_inj__on__eq__iff,axiom,
! [F: produc270694976lle_tv > produc270694976lle_tv,A: set_Pr13445366lle_tv,X: produc270694976lle_tv,Y: produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F @ A )
=> ( ( member813679063lle_tv @ X @ A )
=> ( ( member813679063lle_tv @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_72_inj__on__eq__iff,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A: set_Pr1986765409at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F @ A )
=> ( ( member701585322at_nat @ X @ A )
=> ( ( member701585322at_nat @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_73_inj__on__eq__iff,axiom,
! [F: nat > paraco1605129243lle_tv,A: set_nat,X: nat,Y: nat] :
( ( inj_on1955349912lle_tv @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_74_inj__on__eq__iff,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_75_inj__on__eq__iff,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F @ A )
=> ( ( member266900804lle_tv @ X @ A )
=> ( ( member266900804lle_tv @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_76_assms,axiom,
inj_on_nat_nat @ f @ top_top_set_nat ).
% assms
thf(fact_77_top__set__def,axiom,
( top_to975729152tv_nat
= ( collec848462447tv_nat @ top_to900819589_nat_o ) ) ).
% top_set_def
thf(fact_78_top__set__def,axiom,
( top_to1411004326lle_tv
= ( collec924990229lle_tv @ top_to1889334367e_tv_o ) ) ).
% top_set_def
thf(fact_79_top__set__def,axiom,
( top_to1555900337at_nat
= ( collec7649004at_nat @ top_to632227436_nat_o ) ) ).
% top_set_def
thf(fact_80_top__set__def,axiom,
( top_to1057771083lle_tv
= ( collec1111687558lle_tv @ top_to1299250770e_tv_o ) ) ).
% top_set_def
thf(fact_81_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_82_UNIV__witness,axiom,
? [X3: produc194167194tv_nat] : ( member737151281tv_nat @ X3 @ top_to975729152tv_nat ) ).
% UNIV_witness
thf(fact_83_UNIV__witness,axiom,
? [X3: produc270694976lle_tv] : ( member813679063lle_tv @ X3 @ top_to1411004326lle_tv ) ).
% UNIV_witness
thf(fact_84_UNIV__witness,axiom,
? [X3: product_prod_nat_nat] : ( member701585322at_nat @ X3 @ top_to1555900337at_nat ) ).
% UNIV_witness
thf(fact_85_UNIV__witness,axiom,
? [X3: paraco1605129243lle_tv] : ( member266900804lle_tv @ X3 @ top_to1057771083lle_tv ) ).
% UNIV_witness
thf(fact_86_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_87_UNIV__eq__I,axiom,
! [A: set_Pr1725653840tv_nat] :
( ! [X3: produc194167194tv_nat] : ( member737151281tv_nat @ X3 @ A )
=> ( top_to975729152tv_nat = A ) ) ).
% UNIV_eq_I
thf(fact_88_UNIV__eq__I,axiom,
! [A: set_Pr13445366lle_tv] :
( ! [X3: produc270694976lle_tv] : ( member813679063lle_tv @ X3 @ A )
=> ( top_to1411004326lle_tv = A ) ) ).
% UNIV_eq_I
thf(fact_89_UNIV__eq__I,axiom,
! [A: set_Pr1986765409at_nat] :
( ! [X3: product_prod_nat_nat] : ( member701585322at_nat @ X3 @ A )
=> ( top_to1555900337at_nat = A ) ) ).
% UNIV_eq_I
thf(fact_90_UNIV__eq__I,axiom,
! [A: set_Pa1764573435lle_tv] :
( ! [X3: paraco1605129243lle_tv] : ( member266900804lle_tv @ X3 @ A )
=> ( top_to1057771083lle_tv = A ) ) ).
% UNIV_eq_I
thf(fact_91_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_92_inj__on__inverseI,axiom,
! [A: set_Pr2014382200lle_tv,G: produc1647334210lle_tv > produc1647334210lle_tv,F: produc1647334210lle_tv > produc1647334210lle_tv] :
( ! [X3: produc1647334210lle_tv] :
( ( member1531318361lle_tv @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on2112487900lle_tv @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_93_inj__on__inverseI,axiom,
! [A: set_Pr1725653840tv_nat,G: produc194167194tv_nat > produc194167194tv_nat,F: produc194167194tv_nat > produc194167194tv_nat] :
( ! [X3: produc194167194tv_nat] :
( ( member737151281tv_nat @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on552895452tv_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_94_inj__on__inverseI,axiom,
! [A: set_Pr13445366lle_tv,G: produc270694976lle_tv > produc270694976lle_tv,F: produc270694976lle_tv > produc270694976lle_tv] :
( ! [X3: produc270694976lle_tv] :
( ( member813679063lle_tv @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on1745913142lle_tv @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_95_inj__on__inverseI,axiom,
! [A: set_Pr1986765409at_nat,G: product_prod_nat_nat > product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member701585322at_nat @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on26176689at_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_96_inj__on__inverseI,axiom,
! [A: set_nat,G: paraco1605129243lle_tv > nat,F: nat > paraco1605129243lle_tv] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on1955349912lle_tv @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_97_inj__on__inverseI,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_98_inj__on__inverseI,axiom,
! [A: set_Pa1764573435lle_tv,G: paraco1605129243lle_tv > paraco1605129243lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on772319074lle_tv @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_99_inj__on__contraD,axiom,
! [F: produc1647334210lle_tv > produc1647334210lle_tv,A: set_Pr2014382200lle_tv,X: produc1647334210lle_tv,Y: produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F @ A )
=> ( ( X != Y )
=> ( ( member1531318361lle_tv @ X @ A )
=> ( ( member1531318361lle_tv @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_100_inj__on__contraD,axiom,
! [F: produc194167194tv_nat > produc194167194tv_nat,A: set_Pr1725653840tv_nat,X: produc194167194tv_nat,Y: produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F @ A )
=> ( ( X != Y )
=> ( ( member737151281tv_nat @ X @ A )
=> ( ( member737151281tv_nat @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_101_inj__on__contraD,axiom,
! [F: produc270694976lle_tv > produc270694976lle_tv,A: set_Pr13445366lle_tv,X: produc270694976lle_tv,Y: produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F @ A )
=> ( ( X != Y )
=> ( ( member813679063lle_tv @ X @ A )
=> ( ( member813679063lle_tv @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_102_inj__on__contraD,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A: set_Pr1986765409at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F @ A )
=> ( ( X != Y )
=> ( ( member701585322at_nat @ X @ A )
=> ( ( member701585322at_nat @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_103_inj__on__contraD,axiom,
! [F: nat > paraco1605129243lle_tv,A: set_nat,X: nat,Y: nat] :
( ( inj_on1955349912lle_tv @ F @ A )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_104_inj__on__contraD,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_105_inj__on__contraD,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F @ A )
=> ( ( X != Y )
=> ( ( member266900804lle_tv @ X @ A )
=> ( ( member266900804lle_tv @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_106_top__empty__eq,axiom,
( top_to900819589_nat_o
= ( ^ [X2: produc194167194tv_nat] : ( member737151281tv_nat @ X2 @ top_to975729152tv_nat ) ) ) ).
% top_empty_eq
thf(fact_107_top__empty__eq,axiom,
( top_to1889334367e_tv_o
= ( ^ [X2: produc270694976lle_tv] : ( member813679063lle_tv @ X2 @ top_to1411004326lle_tv ) ) ) ).
% top_empty_eq
thf(fact_108_top__empty__eq,axiom,
( top_to632227436_nat_o
= ( ^ [X2: product_prod_nat_nat] : ( member701585322at_nat @ X2 @ top_to1555900337at_nat ) ) ) ).
% top_empty_eq
thf(fact_109_top__empty__eq,axiom,
( top_to1299250770e_tv_o
= ( ^ [X2: paraco1605129243lle_tv] : ( member266900804lle_tv @ X2 @ top_to1057771083lle_tv ) ) ) ).
% top_empty_eq
thf(fact_110_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_111_the__inv__f__f,axiom,
! [F: produc1647334210lle_tv > produc1647334210lle_tv,X: produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F @ top_to1871154984lle_tv )
=> ( ( the_in427240730lle_tv @ top_to1871154984lle_tv @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_112_the__inv__f__f,axiom,
! [F: paraco1605129243lle_tv > nat,X: paraco1605129243lle_tv] :
( ( inj_on1179712058tv_nat @ F @ top_to1057771083lle_tv )
=> ( ( the_in1509933432tv_nat @ top_to1057771083lle_tv @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_113_the__inv__f__f,axiom,
! [F: nat > paraco1605129243lle_tv,X: nat] :
( ( inj_on1955349912lle_tv @ F @ top_top_set_nat )
=> ( ( the_in138087638lle_tv @ top_top_set_nat @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_114_the__inv__f__f,axiom,
! [F: produc194167194tv_nat > produc194167194tv_nat,X: produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F @ top_to975729152tv_nat )
=> ( ( the_in195660058tv_nat @ top_to975729152tv_nat @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_115_the__inv__f__f,axiom,
! [F: produc270694976lle_tv > produc270694976lle_tv,X: produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F @ top_to1411004326lle_tv )
=> ( ( the_in1388677748lle_tv @ top_to1411004326lle_tv @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_116_the__inv__f__f,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F @ top_to1555900337at_nat )
=> ( ( the_in2068894451at_nat @ top_to1555900337at_nat @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_117_the__inv__f__f,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,X: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F @ top_to1057771083lle_tv )
=> ( ( the_in452116896lle_tv @ top_to1057771083lle_tv @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_118_the__inv__f__f,axiom,
! [F: nat > nat,X: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( the_inv_into_nat_nat @ top_top_set_nat @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_119_strict__mono__on__imp__inj__on,axiom,
! [F: nat > nat,A: set_nat] :
( ( strict2146030304at_nat @ F @ A )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% strict_mono_on_imp_inj_on
thf(fact_120_linorder__injI,axiom,
! [F: nat > paraco1605129243lle_tv] :
( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) )
=> ( inj_on1955349912lle_tv @ F @ top_top_set_nat ) ) ).
% linorder_injI
thf(fact_121_linorder__injI,axiom,
! [F: nat > nat] :
( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% linorder_injI
thf(fact_122_inj__add__left,axiom,
! [A4: nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A4 ) @ top_top_set_nat ) ).
% inj_add_left
thf(fact_123_change__tv_Osimps_I2_J,axiom,
! [F: nat > nat,N: nat] :
( ( paraco1147068288nge_tv @ F @ ( paraco793048302_Indet @ N ) )
= ( paraco793048302_Indet @ ( F @ N ) ) ) ).
% change_tv.simps(2)
thf(fact_124_inj__apsnd,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( inj_on2112487900lle_tv @ ( produc1408152463lle_tv @ F ) @ top_to1871154984lle_tv )
= ( inj_on772319074lle_tv @ F @ top_to1057771083lle_tv ) ) ).
% inj_apsnd
thf(fact_125_inj__apsnd,axiom,
! [F: nat > paraco1605129243lle_tv] :
( ( inj_on967082628lle_tv @ ( produc1804670661lle_tv @ F ) @ top_to975729152tv_nat )
= ( inj_on1955349912lle_tv @ F @ top_top_set_nat ) ) ).
% inj_apsnd
thf(fact_126_inj__apsnd,axiom,
! [F: nat > nat] :
( ( inj_on552895452tv_nat @ ( produc237097339lle_tv @ F ) @ top_to975729152tv_nat )
= ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% inj_apsnd
thf(fact_127_inj__apsnd,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( inj_on1745913142lle_tv @ ( produc2040316391tv_nat @ F ) @ top_to1411004326lle_tv )
= ( inj_on772319074lle_tv @ F @ top_to1057771083lle_tv ) ) ).
% inj_apsnd
thf(fact_128_inj__apsnd,axiom,
! [F: nat > paraco1605129243lle_tv] :
( ( inj_on1840679568lle_tv @ ( produc1608943133tv_nat @ F ) @ top_to1555900337at_nat )
= ( inj_on1955349912lle_tv @ F @ top_top_set_nat ) ) ).
% inj_apsnd
thf(fact_129_inj__apsnd,axiom,
! [F: nat > nat] :
( ( inj_on26176689at_nat @ ( produc1320959600at_nat @ F ) @ top_to1555900337at_nat )
= ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% inj_apsnd
thf(fact_130_inj__image__mem__iff,axiom,
! [F: paraco1605129243lle_tv > nat,A4: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( inj_on1179712058tv_nat @ F @ top_to1057771083lle_tv )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_376325286tv_nat @ F @ A ) )
= ( member266900804lle_tv @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_131_inj__image__mem__iff,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A4: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F @ top_to1057771083lle_tv )
=> ( ( member266900804lle_tv @ ( F @ A4 ) @ ( image_335818702lle_tv @ F @ A ) )
= ( member266900804lle_tv @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_132_inj__image__mem__iff,axiom,
! [F: nat > nat,A4: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_133_inj__image__mem__iff,axiom,
! [F: nat > paraco1605129243lle_tv,A4: nat,A: set_nat] :
( ( inj_on1955349912lle_tv @ F @ top_top_set_nat )
=> ( ( member266900804lle_tv @ ( F @ A4 ) @ ( image_1151963140lle_tv @ F @ A ) )
= ( member_nat @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_134_inj__image__mem__iff,axiom,
! [F: produc194167194tv_nat > nat,A4: produc194167194tv_nat,A: set_Pr1725653840tv_nat] :
( ( inj_on808253149at_nat @ F @ top_to975729152tv_nat )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_1381266801at_nat @ F @ A ) )
= ( member737151281tv_nat @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_135_inj__image__mem__iff,axiom,
! [F: produc194167194tv_nat > paraco1605129243lle_tv,A4: produc194167194tv_nat,A: set_Pr1725653840tv_nat] :
( ( inj_on1612474245lle_tv @ F @ top_to975729152tv_nat )
=> ( ( member266900804lle_tv @ ( F @ A4 ) @ ( image_2019567641lle_tv @ F @ A ) )
= ( member737151281tv_nat @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_136_inj__image__mem__iff,axiom,
! [F: produc270694976lle_tv > nat,A4: produc270694976lle_tv,A: set_Pr13445366lle_tv] :
( ( inj_on936843139tv_nat @ F @ top_to1411004326lle_tv )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_1509856791tv_nat @ F @ A ) )
= ( member813679063lle_tv @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_137_inj__image__mem__iff,axiom,
! [F: produc270694976lle_tv > paraco1605129243lle_tv,A4: produc270694976lle_tv,A: set_Pr13445366lle_tv] :
( ( inj_on394919723lle_tv @ F @ top_to1411004326lle_tv )
=> ( ( member266900804lle_tv @ ( F @ A4 ) @ ( image_802013119lle_tv @ F @ A ) )
= ( member813679063lle_tv @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_138_inj__image__mem__iff,axiom,
! [F: product_prod_nat_nat > nat,A4: product_prod_nat_nat,A: set_Pr1986765409at_nat] :
( ( inj_on642995232at_nat @ F @ top_to1555900337at_nat )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_1397137548at_nat @ F @ A ) )
= ( member701585322at_nat @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_139_inj__image__mem__iff,axiom,
! [F: product_prod_nat_nat > paraco1605129243lle_tv,A4: product_prod_nat_nat,A: set_Pr1986765409at_nat] :
( ( inj_on520021963lle_tv @ F @ top_to1555900337at_nat )
=> ( ( member266900804lle_tv @ ( F @ A4 ) @ ( image_1093035615lle_tv @ F @ A ) )
= ( member701585322at_nat @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_140_inj__image__eq__iff,axiom,
! [F: produc1647334210lle_tv > produc1647334210lle_tv,A: set_Pr2014382200lle_tv,B: set_Pr2014382200lle_tv] :
( ( inj_on2112487900lle_tv @ F @ top_to1871154984lle_tv )
=> ( ( ( image_1221801032lle_tv @ F @ A )
= ( image_1221801032lle_tv @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_141_inj__image__eq__iff,axiom,
! [F: nat > paraco1605129243lle_tv,A: set_nat,B: set_nat] :
( ( inj_on1955349912lle_tv @ F @ top_top_set_nat )
=> ( ( ( image_1151963140lle_tv @ F @ A )
= ( image_1151963140lle_tv @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_142_inj__image__eq__iff,axiom,
! [F: produc194167194tv_nat > produc194167194tv_nat,A: set_Pr1725653840tv_nat,B: set_Pr1725653840tv_nat] :
( ( inj_on552895452tv_nat @ F @ top_to975729152tv_nat )
=> ( ( ( image_1104296520tv_nat @ F @ A )
= ( image_1104296520tv_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_143_inj__image__eq__iff,axiom,
! [F: produc270694976lle_tv > produc270694976lle_tv,A: set_Pr13445366lle_tv,B: set_Pr13445366lle_tv] :
( ( inj_on1745913142lle_tv @ F @ top_to1411004326lle_tv )
=> ( ( ( image_149830562lle_tv @ F @ A )
= ( image_149830562lle_tv @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_144_inj__image__eq__iff,axiom,
! [F: product_prod_nat_nat > produc194167194tv_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( inj_on1764151786tv_nat @ F @ top_to1555900337at_nat )
=> ( ( ( image_1597721174tv_nat @ F @ A )
= ( image_1597721174tv_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_145_inj__image__eq__iff,axiom,
! [F: product_prod_nat_nat > produc270694976lle_tv,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( inj_on1840679568lle_tv @ F @ top_to1555900337at_nat )
=> ( ( ( image_1674248956lle_tv @ F @ A )
= ( image_1674248956lle_tv @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_146_inj__image__eq__iff,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( inj_on26176689at_nat @ F @ top_to1555900337at_nat )
=> ( ( ( image_815163717at_nat @ F @ A )
= ( image_815163717at_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_147_inj__image__eq__iff,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F @ top_to1057771083lle_tv )
=> ( ( ( image_335818702lle_tv @ F @ A )
= ( image_335818702lle_tv @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_148_inj__image__eq__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_149_range__ex1__eq,axiom,
! [F: paraco1605129243lle_tv > nat,B2: nat] :
( ( inj_on1179712058tv_nat @ F @ top_to1057771083lle_tv )
=> ( ( member_nat @ B2 @ ( image_376325286tv_nat @ F @ top_to1057771083lle_tv ) )
= ( ? [X2: paraco1605129243lle_tv] :
( ( B2
= ( F @ X2 ) )
& ! [Y3: paraco1605129243lle_tv] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_150_range__ex1__eq,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,B2: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F @ top_to1057771083lle_tv )
=> ( ( member266900804lle_tv @ B2 @ ( image_335818702lle_tv @ F @ top_to1057771083lle_tv ) )
= ( ? [X2: paraco1605129243lle_tv] :
( ( B2
= ( F @ X2 ) )
& ! [Y3: paraco1605129243lle_tv] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_151_range__ex1__eq,axiom,
! [F: nat > nat,B2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X2: nat] :
( ( B2
= ( F @ X2 ) )
& ! [Y3: nat] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_152_range__ex1__eq,axiom,
! [F: nat > paraco1605129243lle_tv,B2: paraco1605129243lle_tv] :
( ( inj_on1955349912lle_tv @ F @ top_top_set_nat )
=> ( ( member266900804lle_tv @ B2 @ ( image_1151963140lle_tv @ F @ top_top_set_nat ) )
= ( ? [X2: nat] :
( ( B2
= ( F @ X2 ) )
& ! [Y3: nat] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_153_range__ex1__eq,axiom,
! [F: produc194167194tv_nat > nat,B2: nat] :
( ( inj_on808253149at_nat @ F @ top_to975729152tv_nat )
=> ( ( member_nat @ B2 @ ( image_1381266801at_nat @ F @ top_to975729152tv_nat ) )
= ( ? [X2: produc194167194tv_nat] :
( ( B2
= ( F @ X2 ) )
& ! [Y3: produc194167194tv_nat] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_154_range__ex1__eq,axiom,
! [F: produc194167194tv_nat > paraco1605129243lle_tv,B2: paraco1605129243lle_tv] :
( ( inj_on1612474245lle_tv @ F @ top_to975729152tv_nat )
=> ( ( member266900804lle_tv @ B2 @ ( image_2019567641lle_tv @ F @ top_to975729152tv_nat ) )
= ( ? [X2: produc194167194tv_nat] :
( ( B2
= ( F @ X2 ) )
& ! [Y3: produc194167194tv_nat] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_155_range__ex1__eq,axiom,
! [F: produc270694976lle_tv > nat,B2: nat] :
( ( inj_on936843139tv_nat @ F @ top_to1411004326lle_tv )
=> ( ( member_nat @ B2 @ ( image_1509856791tv_nat @ F @ top_to1411004326lle_tv ) )
= ( ? [X2: produc270694976lle_tv] :
( ( B2
= ( F @ X2 ) )
& ! [Y3: produc270694976lle_tv] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_156_range__ex1__eq,axiom,
! [F: produc270694976lle_tv > paraco1605129243lle_tv,B2: paraco1605129243lle_tv] :
( ( inj_on394919723lle_tv @ F @ top_to1411004326lle_tv )
=> ( ( member266900804lle_tv @ B2 @ ( image_802013119lle_tv @ F @ top_to1411004326lle_tv ) )
= ( ? [X2: produc270694976lle_tv] :
( ( B2
= ( F @ X2 ) )
& ! [Y3: produc270694976lle_tv] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_157_range__ex1__eq,axiom,
! [F: product_prod_nat_nat > nat,B2: nat] :
( ( inj_on642995232at_nat @ F @ top_to1555900337at_nat )
=> ( ( member_nat @ B2 @ ( image_1397137548at_nat @ F @ top_to1555900337at_nat ) )
= ( ? [X2: product_prod_nat_nat] :
( ( B2
= ( F @ X2 ) )
& ! [Y3: product_prod_nat_nat] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_158_range__ex1__eq,axiom,
! [F: product_prod_nat_nat > paraco1605129243lle_tv,B2: paraco1605129243lle_tv] :
( ( inj_on520021963lle_tv @ F @ top_to1555900337at_nat )
=> ( ( member266900804lle_tv @ B2 @ ( image_1093035615lle_tv @ F @ top_to1555900337at_nat ) )
= ( ? [X2: product_prod_nat_nat] :
( ( B2
= ( F @ X2 ) )
& ! [Y3: product_prod_nat_nat] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_159_strict__mono__imp__inj__on,axiom,
! [F: nat > nat,A: set_nat] :
( ( order_769474267at_nat @ F )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% strict_mono_imp_inj_on
thf(fact_160_image__eqI,axiom,
! [B2: produc194167194tv_nat,F: product_prod_nat_nat > produc194167194tv_nat,X: product_prod_nat_nat,A: set_Pr1986765409at_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member701585322at_nat @ X @ A )
=> ( member737151281tv_nat @ B2 @ ( image_1597721174tv_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_161_image__eqI,axiom,
! [B2: produc270694976lle_tv,F: product_prod_nat_nat > produc270694976lle_tv,X: product_prod_nat_nat,A: set_Pr1986765409at_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member701585322at_nat @ X @ A )
=> ( member813679063lle_tv @ B2 @ ( image_1674248956lle_tv @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_162_image__eqI,axiom,
! [B2: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,A: set_Pr1986765409at_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member701585322at_nat @ X @ A )
=> ( member701585322at_nat @ B2 @ ( image_815163717at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_163_image__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_164_image__eqI,axiom,
! [B2: paraco1605129243lle_tv,F: nat > paraco1605129243lle_tv,X: nat,A: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member266900804lle_tv @ B2 @ ( image_1151963140lle_tv @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_165_image__eqI,axiom,
! [B2: nat,F: paraco1605129243lle_tv > nat,X: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( B2
= ( F @ X ) )
=> ( ( member266900804lle_tv @ X @ A )
=> ( member_nat @ B2 @ ( image_376325286tv_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_166_image__eqI,axiom,
! [B2: paraco1605129243lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv,X: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( B2
= ( F @ X ) )
=> ( ( member266900804lle_tv @ X @ A )
=> ( member266900804lle_tv @ B2 @ ( image_335818702lle_tv @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_167_tv_Oinject_I2_J,axiom,
! [X22: nat,Y22: nat] :
( ( ( paraco793048302_Indet @ X22 )
= ( paraco793048302_Indet @ Y22 ) )
= ( X22 = Y22 ) ) ).
% tv.inject(2)
thf(fact_168_top1I,axiom,
! [X: paraco1605129243lle_tv] : ( top_to1299250770e_tv_o @ X ) ).
% top1I
thf(fact_169_top1I,axiom,
! [X: nat] : ( top_top_nat_o @ X ) ).
% top1I
thf(fact_170_the__inv__into__onto,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F @ A )
=> ( ( image_335818702lle_tv @ ( the_in452116896lle_tv @ A @ F ) @ ( image_335818702lle_tv @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_171_the__inv__into__onto,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_172_the__inv__into__onto,axiom,
! [F: nat > paraco1605129243lle_tv,A: set_nat] :
( ( inj_on1955349912lle_tv @ F @ A )
=> ( ( image_376325286tv_nat @ ( the_in138087638lle_tv @ A @ F ) @ ( image_1151963140lle_tv @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_173_the__inv__into__onto,axiom,
! [F: paraco1605129243lle_tv > nat,A: set_Pa1764573435lle_tv] :
( ( inj_on1179712058tv_nat @ F @ A )
=> ( ( image_1151963140lle_tv @ ( the_in1509933432tv_nat @ A @ F ) @ ( image_376325286tv_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_174_the__inv__into__onto,axiom,
! [F: product_prod_nat_nat > produc194167194tv_nat,A: set_Pr1986765409at_nat] :
( ( inj_on1764151786tv_nat @ F @ A )
=> ( ( image_1805124128at_nat @ ( the_in667485224tv_nat @ A @ F ) @ ( image_1597721174tv_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_175_the__inv__into__onto,axiom,
! [F: product_prod_nat_nat > produc270694976lle_tv,A: set_Pr1986765409at_nat] :
( ( inj_on1840679568lle_tv @ F @ A )
=> ( ( image_1787705850at_nat @ ( the_in744013006lle_tv @ A @ F ) @ ( image_1674248956lle_tv @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_176_the__inv__into__onto,axiom,
! [F: produc194167194tv_nat > product_prod_nat_nat,A: set_Pr1725653840tv_nat] :
( ( inj_on1971554740at_nat @ F @ A )
=> ( ( image_1597721174tv_nat @ ( the_in874888178at_nat @ A @ F ) @ ( image_1805124128at_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_177_the__inv__into__onto,axiom,
! [F: produc270694976lle_tv > product_prod_nat_nat,A: set_Pr13445366lle_tv] :
( ( inj_on1954136462at_nat @ F @ A )
=> ( ( image_1674248956lle_tv @ ( the_in857469900at_nat @ A @ F ) @ ( image_1787705850at_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_178_the__inv__into__onto,axiom,
! [F: produc1647334210lle_tv > produc1647334210lle_tv,A: set_Pr2014382200lle_tv] :
( ( inj_on2112487900lle_tv @ F @ A )
=> ( ( image_1221801032lle_tv @ ( the_in427240730lle_tv @ A @ F ) @ ( image_1221801032lle_tv @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_179_the__inv__into__onto,axiom,
! [F: produc194167194tv_nat > produc194167194tv_nat,A: set_Pr1725653840tv_nat] :
( ( inj_on552895452tv_nat @ F @ A )
=> ( ( image_1104296520tv_nat @ ( the_in195660058tv_nat @ A @ F ) @ ( image_1104296520tv_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_180_imageI,axiom,
! [X: product_prod_nat_nat,A: set_Pr1986765409at_nat,F: product_prod_nat_nat > produc194167194tv_nat] :
( ( member701585322at_nat @ X @ A )
=> ( member737151281tv_nat @ ( F @ X ) @ ( image_1597721174tv_nat @ F @ A ) ) ) ).
% imageI
thf(fact_181_imageI,axiom,
! [X: product_prod_nat_nat,A: set_Pr1986765409at_nat,F: product_prod_nat_nat > produc270694976lle_tv] :
( ( member701585322at_nat @ X @ A )
=> ( member813679063lle_tv @ ( F @ X ) @ ( image_1674248956lle_tv @ F @ A ) ) ) ).
% imageI
thf(fact_182_imageI,axiom,
! [X: product_prod_nat_nat,A: set_Pr1986765409at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
( ( member701585322at_nat @ X @ A )
=> ( member701585322at_nat @ ( F @ X ) @ ( image_815163717at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_183_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_184_imageI,axiom,
! [X: nat,A: set_nat,F: nat > paraco1605129243lle_tv] :
( ( member_nat @ X @ A )
=> ( member266900804lle_tv @ ( F @ X ) @ ( image_1151963140lle_tv @ F @ A ) ) ) ).
% imageI
thf(fact_185_imageI,axiom,
! [X: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > nat] :
( ( member266900804lle_tv @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_376325286tv_nat @ F @ A ) ) ) ).
% imageI
thf(fact_186_imageI,axiom,
! [X: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ A )
=> ( member266900804lle_tv @ ( F @ X ) @ ( image_335818702lle_tv @ F @ A ) ) ) ).
% imageI
thf(fact_187_image__iff,axiom,
! [Z: produc194167194tv_nat,F: product_prod_nat_nat > produc194167194tv_nat,A: set_Pr1986765409at_nat] :
( ( member737151281tv_nat @ Z @ ( image_1597721174tv_nat @ F @ A ) )
= ( ? [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_188_image__iff,axiom,
! [Z: produc270694976lle_tv,F: product_prod_nat_nat > produc270694976lle_tv,A: set_Pr1986765409at_nat] :
( ( member813679063lle_tv @ Z @ ( image_1674248956lle_tv @ F @ A ) )
= ( ? [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_189_image__iff,axiom,
! [Z: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat,A: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ Z @ ( image_815163717at_nat @ F @ A ) )
= ( ? [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_190_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_191_image__iff,axiom,
! [Z: paraco1605129243lle_tv,F: nat > paraco1605129243lle_tv,A: set_nat] :
( ( member266900804lle_tv @ Z @ ( image_1151963140lle_tv @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_192_bex__imageD,axiom,
! [F: product_prod_nat_nat > produc194167194tv_nat,A: set_Pr1986765409at_nat,P: produc194167194tv_nat > $o] :
( ? [X4: produc194167194tv_nat] :
( ( member737151281tv_nat @ X4 @ ( image_1597721174tv_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: product_prod_nat_nat] :
( ( member701585322at_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_193_bex__imageD,axiom,
! [F: product_prod_nat_nat > produc270694976lle_tv,A: set_Pr1986765409at_nat,P: produc270694976lle_tv > $o] :
( ? [X4: produc270694976lle_tv] :
( ( member813679063lle_tv @ X4 @ ( image_1674248956lle_tv @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: product_prod_nat_nat] :
( ( member701585322at_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_194_bex__imageD,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A: set_Pr1986765409at_nat,P: product_prod_nat_nat > $o] :
( ? [X4: product_prod_nat_nat] :
( ( member701585322at_nat @ X4 @ ( image_815163717at_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: product_prod_nat_nat] :
( ( member701585322at_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_195_bex__imageD,axiom,
! [F: nat > paraco1605129243lle_tv,A: set_nat,P: paraco1605129243lle_tv > $o] :
( ? [X4: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X4 @ ( image_1151963140lle_tv @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_196_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_197_image__cong,axiom,
! [M: set_Pr1986765409at_nat,N2: set_Pr1986765409at_nat,F: product_prod_nat_nat > produc194167194tv_nat,G: product_prod_nat_nat > produc194167194tv_nat] :
( ( M = N2 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member701585322at_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_1597721174tv_nat @ F @ M )
= ( image_1597721174tv_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_198_image__cong,axiom,
! [M: set_Pr1986765409at_nat,N2: set_Pr1986765409at_nat,F: product_prod_nat_nat > produc270694976lle_tv,G: product_prod_nat_nat > produc270694976lle_tv] :
( ( M = N2 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member701585322at_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_1674248956lle_tv @ F @ M )
= ( image_1674248956lle_tv @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_199_image__cong,axiom,
! [M: set_Pr1986765409at_nat,N2: set_Pr1986765409at_nat,F: product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat] :
( ( M = N2 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member701585322at_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_815163717at_nat @ F @ M )
= ( image_815163717at_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_200_image__cong,axiom,
! [M: set_nat,N2: set_nat,F: nat > paraco1605129243lle_tv,G: nat > paraco1605129243lle_tv] :
( ( M = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_1151963140lle_tv @ F @ M )
= ( image_1151963140lle_tv @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_201_image__cong,axiom,
! [M: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_202_ball__imageD,axiom,
! [F: product_prod_nat_nat > produc194167194tv_nat,A: set_Pr1986765409at_nat,P: produc194167194tv_nat > $o] :
( ! [X3: produc194167194tv_nat] :
( ( member737151281tv_nat @ X3 @ ( image_1597721174tv_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: product_prod_nat_nat] :
( ( member701585322at_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_203_ball__imageD,axiom,
! [F: product_prod_nat_nat > produc270694976lle_tv,A: set_Pr1986765409at_nat,P: produc270694976lle_tv > $o] :
( ! [X3: produc270694976lle_tv] :
( ( member813679063lle_tv @ X3 @ ( image_1674248956lle_tv @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: product_prod_nat_nat] :
( ( member701585322at_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_204_ball__imageD,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A: set_Pr1986765409at_nat,P: product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat] :
( ( member701585322at_nat @ X3 @ ( image_815163717at_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: product_prod_nat_nat] :
( ( member701585322at_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_205_ball__imageD,axiom,
! [F: nat > paraco1605129243lle_tv,A: set_nat,P: paraco1605129243lle_tv > $o] :
( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ ( image_1151963140lle_tv @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_206_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_207_rev__image__eqI,axiom,
! [X: product_prod_nat_nat,A: set_Pr1986765409at_nat,B2: produc194167194tv_nat,F: product_prod_nat_nat > produc194167194tv_nat] :
( ( member701585322at_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member737151281tv_nat @ B2 @ ( image_1597721174tv_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_208_rev__image__eqI,axiom,
! [X: product_prod_nat_nat,A: set_Pr1986765409at_nat,B2: produc270694976lle_tv,F: product_prod_nat_nat > produc270694976lle_tv] :
( ( member701585322at_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member813679063lle_tv @ B2 @ ( image_1674248956lle_tv @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_209_rev__image__eqI,axiom,
! [X: product_prod_nat_nat,A: set_Pr1986765409at_nat,B2: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
( ( member701585322at_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member701585322at_nat @ B2 @ ( image_815163717at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_210_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_211_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: paraco1605129243lle_tv,F: nat > paraco1605129243lle_tv] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member266900804lle_tv @ B2 @ ( image_1151963140lle_tv @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_212_rev__image__eqI,axiom,
! [X: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,B2: nat,F: paraco1605129243lle_tv > nat] :
( ( member266900804lle_tv @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_376325286tv_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_213_rev__image__eqI,axiom,
! [X: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,B2: paraco1605129243lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member266900804lle_tv @ B2 @ ( image_335818702lle_tv @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_214_strict__mono__onD,axiom,
! [F: nat > nat,A: set_nat,R: nat,S: nat] :
( ( strict2146030304at_nat @ F @ A )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S @ A )
=> ( ( ord_less_nat @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_215_strict__mono__onD,axiom,
! [F: nat > set_nat,A: set_nat,R: nat,S: nat] :
( ( strict1310998166et_nat @ F @ A )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S @ A )
=> ( ( ord_less_nat @ R @ S )
=> ( ord_less_set_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_216_strict__mono__onD,axiom,
! [F: nat > set_Pa1764573435lle_tv,A: set_nat,R: nat,S: nat] :
( ( strict1658340843lle_tv @ F @ A )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S @ A )
=> ( ( ord_less_nat @ R @ S )
=> ( ord_le727057319lle_tv @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_217_strict__mono__onD,axiom,
! [F: set_nat > nat,A: set_set_nat,R: set_nat,S: set_nat] :
( ( strict1802283670at_nat @ F @ A )
=> ( ( member_set_nat @ R @ A )
=> ( ( member_set_nat @ S @ A )
=> ( ( ord_less_set_nat @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_218_strict__mono__onD,axiom,
! [F: set_nat > set_nat,A: set_set_nat,R: set_nat,S: set_nat] :
( ( strict473775692et_nat @ F @ A )
=> ( ( member_set_nat @ R @ A )
=> ( ( member_set_nat @ S @ A )
=> ( ( ord_less_set_nat @ R @ S )
=> ( ord_less_set_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_219_strict__mono__onD,axiom,
! [F: set_nat > set_Pa1764573435lle_tv,A: set_set_nat,R: set_nat,S: set_nat] :
( ( strict172559285lle_tv @ F @ A )
=> ( ( member_set_nat @ R @ A )
=> ( ( member_set_nat @ S @ A )
=> ( ( ord_less_set_nat @ R @ S )
=> ( ord_le727057319lle_tv @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_220_strict__mono__onD,axiom,
! [F: set_Pa1764573435lle_tv > nat,A: set_se800614875lle_tv,R: set_Pa1764573435lle_tv,S: set_Pa1764573435lle_tv] :
( ( strict1125660301tv_nat @ F @ A )
=> ( ( member1875402276lle_tv @ R @ A )
=> ( ( member1875402276lle_tv @ S @ A )
=> ( ( ord_le727057319lle_tv @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_221_strict__mono__onD,axiom,
! [F: set_Pa1764573435lle_tv > set_nat,A: set_se800614875lle_tv,R: set_Pa1764573435lle_tv,S: set_Pa1764573435lle_tv] :
( ( strict1455779395et_nat @ F @ A )
=> ( ( member1875402276lle_tv @ R @ A )
=> ( ( member1875402276lle_tv @ S @ A )
=> ( ( ord_le727057319lle_tv @ R @ S )
=> ( ord_less_set_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_222_strict__mono__onD,axiom,
! [F: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv,A: set_se800614875lle_tv,R: set_Pa1764573435lle_tv,S: set_Pa1764573435lle_tv] :
( ( strict592849515lle_tv @ F @ A )
=> ( ( member1875402276lle_tv @ R @ A )
=> ( ( member1875402276lle_tv @ S @ A )
=> ( ( ord_le727057319lle_tv @ R @ S )
=> ( ord_le727057319lle_tv @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_223_strict__mono__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [R2: nat,S2: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_nat @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( strict2146030304at_nat @ F @ A ) ) ).
% strict_mono_onI
thf(fact_224_strict__mono__onI,axiom,
! [A: set_nat,F: nat > set_nat] :
( ! [R2: nat,S2: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_nat @ R2 @ S2 )
=> ( ord_less_set_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( strict1310998166et_nat @ F @ A ) ) ).
% strict_mono_onI
thf(fact_225_strict__mono__onI,axiom,
! [A: set_nat,F: nat > set_Pa1764573435lle_tv] :
( ! [R2: nat,S2: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_nat @ R2 @ S2 )
=> ( ord_le727057319lle_tv @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( strict1658340843lle_tv @ F @ A ) ) ).
% strict_mono_onI
thf(fact_226_strict__mono__onI,axiom,
! [A: set_set_nat,F: set_nat > nat] :
( ! [R2: set_nat,S2: set_nat] :
( ( member_set_nat @ R2 @ A )
=> ( ( member_set_nat @ S2 @ A )
=> ( ( ord_less_set_nat @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( strict1802283670at_nat @ F @ A ) ) ).
% strict_mono_onI
thf(fact_227_strict__mono__onI,axiom,
! [A: set_set_nat,F: set_nat > set_nat] :
( ! [R2: set_nat,S2: set_nat] :
( ( member_set_nat @ R2 @ A )
=> ( ( member_set_nat @ S2 @ A )
=> ( ( ord_less_set_nat @ R2 @ S2 )
=> ( ord_less_set_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( strict473775692et_nat @ F @ A ) ) ).
% strict_mono_onI
thf(fact_228_strict__mono__onI,axiom,
! [A: set_set_nat,F: set_nat > set_Pa1764573435lle_tv] :
( ! [R2: set_nat,S2: set_nat] :
( ( member_set_nat @ R2 @ A )
=> ( ( member_set_nat @ S2 @ A )
=> ( ( ord_less_set_nat @ R2 @ S2 )
=> ( ord_le727057319lle_tv @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( strict172559285lle_tv @ F @ A ) ) ).
% strict_mono_onI
thf(fact_229_strict__mono__onI,axiom,
! [A: set_se800614875lle_tv,F: set_Pa1764573435lle_tv > nat] :
( ! [R2: set_Pa1764573435lle_tv,S2: set_Pa1764573435lle_tv] :
( ( member1875402276lle_tv @ R2 @ A )
=> ( ( member1875402276lle_tv @ S2 @ A )
=> ( ( ord_le727057319lle_tv @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( strict1125660301tv_nat @ F @ A ) ) ).
% strict_mono_onI
thf(fact_230_strict__mono__onI,axiom,
! [A: set_se800614875lle_tv,F: set_Pa1764573435lle_tv > set_nat] :
( ! [R2: set_Pa1764573435lle_tv,S2: set_Pa1764573435lle_tv] :
( ( member1875402276lle_tv @ R2 @ A )
=> ( ( member1875402276lle_tv @ S2 @ A )
=> ( ( ord_le727057319lle_tv @ R2 @ S2 )
=> ( ord_less_set_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( strict1455779395et_nat @ F @ A ) ) ).
% strict_mono_onI
thf(fact_231_strict__mono__onI,axiom,
! [A: set_se800614875lle_tv,F: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv] :
( ! [R2: set_Pa1764573435lle_tv,S2: set_Pa1764573435lle_tv] :
( ( member1875402276lle_tv @ R2 @ A )
=> ( ( member1875402276lle_tv @ S2 @ A )
=> ( ( ord_le727057319lle_tv @ R2 @ S2 )
=> ( ord_le727057319lle_tv @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( strict592849515lle_tv @ F @ A ) ) ).
% strict_mono_onI
thf(fact_232_strict__mono__on__def,axiom,
( strict2146030304at_nat
= ( ^ [F2: nat > nat,A2: set_nat] :
! [R3: nat,S3: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S3 @ A2 )
& ( ord_less_nat @ R3 @ S3 ) )
=> ( ord_less_nat @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_233_strict__mono__on__def,axiom,
( strict1310998166et_nat
= ( ^ [F2: nat > set_nat,A2: set_nat] :
! [R3: nat,S3: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S3 @ A2 )
& ( ord_less_nat @ R3 @ S3 ) )
=> ( ord_less_set_nat @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_234_strict__mono__on__def,axiom,
( strict1658340843lle_tv
= ( ^ [F2: nat > set_Pa1764573435lle_tv,A2: set_nat] :
! [R3: nat,S3: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S3 @ A2 )
& ( ord_less_nat @ R3 @ S3 ) )
=> ( ord_le727057319lle_tv @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_235_strict__mono__on__def,axiom,
( strict1802283670at_nat
= ( ^ [F2: set_nat > nat,A2: set_set_nat] :
! [R3: set_nat,S3: set_nat] :
( ( ( member_set_nat @ R3 @ A2 )
& ( member_set_nat @ S3 @ A2 )
& ( ord_less_set_nat @ R3 @ S3 ) )
=> ( ord_less_nat @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_236_strict__mono__on__def,axiom,
( strict473775692et_nat
= ( ^ [F2: set_nat > set_nat,A2: set_set_nat] :
! [R3: set_nat,S3: set_nat] :
( ( ( member_set_nat @ R3 @ A2 )
& ( member_set_nat @ S3 @ A2 )
& ( ord_less_set_nat @ R3 @ S3 ) )
=> ( ord_less_set_nat @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_237_strict__mono__on__def,axiom,
( strict172559285lle_tv
= ( ^ [F2: set_nat > set_Pa1764573435lle_tv,A2: set_set_nat] :
! [R3: set_nat,S3: set_nat] :
( ( ( member_set_nat @ R3 @ A2 )
& ( member_set_nat @ S3 @ A2 )
& ( ord_less_set_nat @ R3 @ S3 ) )
=> ( ord_le727057319lle_tv @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_238_strict__mono__on__def,axiom,
( strict1125660301tv_nat
= ( ^ [F2: set_Pa1764573435lle_tv > nat,A2: set_se800614875lle_tv] :
! [R3: set_Pa1764573435lle_tv,S3: set_Pa1764573435lle_tv] :
( ( ( member1875402276lle_tv @ R3 @ A2 )
& ( member1875402276lle_tv @ S3 @ A2 )
& ( ord_le727057319lle_tv @ R3 @ S3 ) )
=> ( ord_less_nat @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_239_strict__mono__on__def,axiom,
( strict1455779395et_nat
= ( ^ [F2: set_Pa1764573435lle_tv > set_nat,A2: set_se800614875lle_tv] :
! [R3: set_Pa1764573435lle_tv,S3: set_Pa1764573435lle_tv] :
( ( ( member1875402276lle_tv @ R3 @ A2 )
& ( member1875402276lle_tv @ S3 @ A2 )
& ( ord_le727057319lle_tv @ R3 @ S3 ) )
=> ( ord_less_set_nat @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_240_strict__mono__on__def,axiom,
( strict592849515lle_tv
= ( ^ [F2: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv,A2: set_se800614875lle_tv] :
! [R3: set_Pa1764573435lle_tv,S3: set_Pa1764573435lle_tv] :
( ( ( member1875402276lle_tv @ R3 @ A2 )
& ( member1875402276lle_tv @ S3 @ A2 )
& ( ord_le727057319lle_tv @ R3 @ S3 ) )
=> ( ord_le727057319lle_tv @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_241_ord__eq__less__subst,axiom,
! [A4: nat,F: nat > nat,B2: nat,C: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_242_ord__eq__less__subst,axiom,
! [A4: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_243_ord__eq__less__subst,axiom,
! [A4: set_Pa1764573435lle_tv,F: nat > set_Pa1764573435lle_tv,B2: nat,C: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le727057319lle_tv @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le727057319lle_tv @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_244_ord__eq__less__subst,axiom,
! [A4: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_245_ord__eq__less__subst,axiom,
! [A4: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_246_ord__eq__less__subst,axiom,
! [A4: set_Pa1764573435lle_tv,F: set_nat > set_Pa1764573435lle_tv,B2: set_nat,C: set_nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X3 @ Y2 )
=> ( ord_le727057319lle_tv @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le727057319lle_tv @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_247_ord__eq__less__subst,axiom,
! [A4: nat,F: set_Pa1764573435lle_tv > nat,B2: set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le727057319lle_tv @ B2 @ C )
=> ( ! [X3: set_Pa1764573435lle_tv,Y2: set_Pa1764573435lle_tv] :
( ( ord_le727057319lle_tv @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_248_ord__eq__less__subst,axiom,
! [A4: set_nat,F: set_Pa1764573435lle_tv > set_nat,B2: set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le727057319lle_tv @ B2 @ C )
=> ( ! [X3: set_Pa1764573435lle_tv,Y2: set_Pa1764573435lle_tv] :
( ( ord_le727057319lle_tv @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_249_ord__eq__less__subst,axiom,
! [A4: set_Pa1764573435lle_tv,F: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv,B2: set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le727057319lle_tv @ B2 @ C )
=> ( ! [X3: set_Pa1764573435lle_tv,Y2: set_Pa1764573435lle_tv] :
( ( ord_le727057319lle_tv @ X3 @ Y2 )
=> ( ord_le727057319lle_tv @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le727057319lle_tv @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_250_mem__Collect__eq,axiom,
! [A4: nat,P: nat > $o] :
( ( member_nat @ A4 @ ( collect_nat @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_251_mem__Collect__eq,axiom,
! [A4: paraco1605129243lle_tv,P: paraco1605129243lle_tv > $o] :
( ( member266900804lle_tv @ A4 @ ( collec1111687558lle_tv @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_252_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_253_Collect__mem__eq,axiom,
! [A: set_Pa1764573435lle_tv] :
( ( collec1111687558lle_tv
@ ^ [X2: paraco1605129243lle_tv] : ( member266900804lle_tv @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_254_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_255_Collect__cong,axiom,
! [P: paraco1605129243lle_tv > $o,Q: paraco1605129243lle_tv > $o] :
( ! [X3: paraco1605129243lle_tv] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec1111687558lle_tv @ P )
= ( collec1111687558lle_tv @ Q ) ) ) ).
% Collect_cong
thf(fact_256_inj__on__strict__subset,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,B: set_Pa1764573435lle_tv,A: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F @ B )
=> ( ( ord_le727057319lle_tv @ A @ B )
=> ( ord_le727057319lle_tv @ ( image_335818702lle_tv @ F @ A ) @ ( image_335818702lle_tv @ F @ B ) ) ) ) ).
% inj_on_strict_subset
thf(fact_257_inj__on__strict__subset,axiom,
! [F: nat > nat,B: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( ord_less_set_nat @ A @ B )
=> ( ord_less_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ) ).
% inj_on_strict_subset
thf(fact_258_inj__on__the__inv__into,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F @ A )
=> ( inj_on772319074lle_tv @ ( the_in452116896lle_tv @ A @ F ) @ ( image_335818702lle_tv @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_259_inj__on__the__inv__into,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_260_f__the__inv__into__f,axiom,
! [F: nat > nat,A: set_nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ Y @ ( image_nat_nat @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_nat @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_261_f__the__inv__into__f,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F @ A )
=> ( ( member266900804lle_tv @ Y @ ( image_335818702lle_tv @ F @ A ) )
=> ( ( F @ ( the_in452116896lle_tv @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_262_strict__mono__inv,axiom,
! [F: nat > nat,G: nat > nat] :
( ( order_769474267at_nat @ F )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( order_769474267at_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_263_surjD,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( ( image_335818702lle_tv @ F @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ? [X3: paraco1605129243lle_tv] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_264_surjD,axiom,
! [F: paraco1605129243lle_tv > nat,Y: nat] :
( ( ( image_376325286tv_nat @ F @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ? [X3: paraco1605129243lle_tv] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_265_surjD,axiom,
! [F: nat > paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( ( image_1151963140lle_tv @ F @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_266_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_267_surjE,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( ( image_335818702lle_tv @ F @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ~ ! [X3: paraco1605129243lle_tv] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_268_surjE,axiom,
! [F: paraco1605129243lle_tv > nat,Y: nat] :
( ( ( image_376325286tv_nat @ F @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ~ ! [X3: paraco1605129243lle_tv] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_269_surjE,axiom,
! [F: nat > paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( ( image_1151963140lle_tv @ F @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_270_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_271_surjI,axiom,
! [G: paraco1605129243lle_tv > paraco1605129243lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [X3: paraco1605129243lle_tv] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_335818702lle_tv @ G @ top_to1057771083lle_tv )
= top_to1057771083lle_tv ) ) ).
% surjI
thf(fact_272_surjI,axiom,
! [G: paraco1605129243lle_tv > nat,F: nat > paraco1605129243lle_tv] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_376325286tv_nat @ G @ top_to1057771083lle_tv )
= top_top_set_nat ) ) ).
% surjI
thf(fact_273_surjI,axiom,
! [G: nat > paraco1605129243lle_tv,F: paraco1605129243lle_tv > nat] :
( ! [X3: paraco1605129243lle_tv] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_1151963140lle_tv @ G @ top_top_set_nat )
= top_to1057771083lle_tv ) ) ).
% surjI
thf(fact_274_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_275_rangeI,axiom,
! [F: paraco1605129243lle_tv > nat,X: paraco1605129243lle_tv] : ( member_nat @ ( F @ X ) @ ( image_376325286tv_nat @ F @ top_to1057771083lle_tv ) ) ).
% rangeI
thf(fact_276_rangeI,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,X: paraco1605129243lle_tv] : ( member266900804lle_tv @ ( F @ X ) @ ( image_335818702lle_tv @ F @ top_to1057771083lle_tv ) ) ).
% rangeI
thf(fact_277_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_278_rangeI,axiom,
! [F: nat > paraco1605129243lle_tv,X: nat] : ( member266900804lle_tv @ ( F @ X ) @ ( image_1151963140lle_tv @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_279_surj__def,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( ( image_335818702lle_tv @ F @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
= ( ! [Y3: paraco1605129243lle_tv] :
? [X2: paraco1605129243lle_tv] :
( Y3
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_280_surj__def,axiom,
! [F: paraco1605129243lle_tv > nat] :
( ( ( image_376325286tv_nat @ F @ top_to1057771083lle_tv )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X2: paraco1605129243lle_tv] :
( Y3
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_281_surj__def,axiom,
! [F: nat > paraco1605129243lle_tv] :
( ( ( image_1151963140lle_tv @ F @ top_top_set_nat )
= top_to1057771083lle_tv )
= ( ! [Y3: paraco1605129243lle_tv] :
? [X2: nat] :
( Y3
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_282_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X2: nat] :
( Y3
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_283_range__eqI,axiom,
! [B2: nat,F: paraco1605129243lle_tv > nat,X: paraco1605129243lle_tv] :
( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_376325286tv_nat @ F @ top_to1057771083lle_tv ) ) ) ).
% range_eqI
thf(fact_284_range__eqI,axiom,
! [B2: paraco1605129243lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv,X: paraco1605129243lle_tv] :
( ( B2
= ( F @ X ) )
=> ( member266900804lle_tv @ B2 @ ( image_335818702lle_tv @ F @ top_to1057771083lle_tv ) ) ) ).
% range_eqI
thf(fact_285_range__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat] :
( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_286_range__eqI,axiom,
! [B2: paraco1605129243lle_tv,F: nat > paraco1605129243lle_tv,X: nat] :
( ( B2
= ( F @ X ) )
=> ( member266900804lle_tv @ B2 @ ( image_1151963140lle_tv @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_287_inj__on__image__iff,axiom,
! [A: set_Pa1764573435lle_tv,G: paraco1605129243lle_tv > paraco1605129243lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ A )
=> ! [Xa: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ Xa @ A )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on772319074lle_tv @ F @ A )
=> ( ( inj_on772319074lle_tv @ G @ ( image_335818702lle_tv @ F @ A ) )
= ( inj_on772319074lle_tv @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_288_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_289_top_Oextremum__strict,axiom,
! [A4: set_Pa1764573435lle_tv] :
~ ( ord_le727057319lle_tv @ top_to1057771083lle_tv @ A4 ) ).
% top.extremum_strict
thf(fact_290_top_Oextremum__strict,axiom,
! [A4: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A4 ) ).
% top.extremum_strict
thf(fact_291_top_Onot__eq__extremum,axiom,
! [A4: set_Pa1764573435lle_tv] :
( ( A4 != top_to1057771083lle_tv )
= ( ord_le727057319lle_tv @ A4 @ top_to1057771083lle_tv ) ) ).
% top.not_eq_extremum
thf(fact_292_top_Onot__eq__extremum,axiom,
! [A4: set_nat] :
( ( A4 != top_top_set_nat )
= ( ord_less_set_nat @ A4 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_293_inj__on__add,axiom,
! [A4: nat,A: set_nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A4 ) @ A ) ).
% inj_on_add
thf(fact_294_the__inv__into__f__f,axiom,
! [F: nat > nat,A: set_nat,X: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_295_the__inv__into__f__f,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F @ A )
=> ( ( member266900804lle_tv @ X @ A )
=> ( ( the_in452116896lle_tv @ A @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_296_the__inv__into__f__eq,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X )
= Y )
=> ( ( member_nat @ X @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_297_the__inv__into__f__eq,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F @ A )
=> ( ( ( F @ X )
= Y )
=> ( ( member266900804lle_tv @ X @ A )
=> ( ( the_in452116896lle_tv @ A @ F @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_298_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_299_psubsetD,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv,C: paraco1605129243lle_tv] :
( ( ord_le727057319lle_tv @ A @ B )
=> ( ( member266900804lle_tv @ C @ A )
=> ( member266900804lle_tv @ C @ B ) ) ) ).
% psubsetD
thf(fact_300_the__inv__into__into,axiom,
! [F: nat > nat,A: set_nat,X: nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( member_nat @ ( the_inv_into_nat_nat @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_301_the__inv__into__into,axiom,
! [F: paraco1605129243lle_tv > nat,A: set_Pa1764573435lle_tv,X: nat,B: set_Pa1764573435lle_tv] :
( ( inj_on1179712058tv_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_376325286tv_nat @ F @ A ) )
=> ( ( ord_le241064091lle_tv @ A @ B )
=> ( member266900804lle_tv @ ( the_in1509933432tv_nat @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_302_the__inv__into__into,axiom,
! [F: nat > paraco1605129243lle_tv,A: set_nat,X: paraco1605129243lle_tv,B: set_nat] :
( ( inj_on1955349912lle_tv @ F @ A )
=> ( ( member266900804lle_tv @ X @ ( image_1151963140lle_tv @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( member_nat @ ( the_in138087638lle_tv @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_303_the__inv__into__into,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv,B: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F @ A )
=> ( ( member266900804lle_tv @ X @ ( image_335818702lle_tv @ F @ A ) )
=> ( ( ord_le241064091lle_tv @ A @ B )
=> ( member266900804lle_tv @ ( the_in452116896lle_tv @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_304_prod_Oinj__map,axiom,
! [F1: paraco1605129243lle_tv > paraco1605129243lle_tv,F22: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F1 @ top_to1057771083lle_tv )
=> ( ( inj_on772319074lle_tv @ F22 @ top_to1057771083lle_tv )
=> ( inj_on2112487900lle_tv @ ( produc1345917071lle_tv @ F1 @ F22 ) @ top_to1871154984lle_tv ) ) ) ).
% prod.inj_map
thf(fact_305_prod_Oinj__map,axiom,
! [F1: paraco1605129243lle_tv > paraco1605129243lle_tv,F22: nat > nat] :
( ( inj_on772319074lle_tv @ F1 @ top_to1057771083lle_tv )
=> ( ( inj_on_nat_nat @ F22 @ top_top_set_nat )
=> ( inj_on552895452tv_nat @ ( produc42536895at_nat @ F1 @ F22 ) @ top_to975729152tv_nat ) ) ) ).
% prod.inj_map
thf(fact_306_prod_Oinj__map,axiom,
! [F1: nat > nat,F22: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( inj_on_nat_nat @ F1 @ top_top_set_nat )
=> ( ( inj_on772319074lle_tv @ F22 @ top_to1057771083lle_tv )
=> ( inj_on1745913142lle_tv @ ( produc2101297531lle_tv @ F1 @ F22 ) @ top_to1411004326lle_tv ) ) ) ).
% prod.inj_map
thf(fact_307_prod_Oinj__map,axiom,
! [F1: nat > nat,F22: nat > nat] :
( ( inj_on_nat_nat @ F1 @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F22 @ top_top_set_nat )
=> ( inj_on26176689at_nat @ ( produc505105722at_nat @ F1 @ F22 ) @ top_to1555900337at_nat ) ) ) ).
% prod.inj_map
thf(fact_308_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_309_subsetI,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ A )
=> ( member266900804lle_tv @ X3 @ B ) )
=> ( ord_le241064091lle_tv @ A @ B ) ) ).
% subsetI
thf(fact_310_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_311_image__subsetI,axiom,
! [A: set_nat,F: nat > paraco1605129243lle_tv,B: set_Pa1764573435lle_tv] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member266900804lle_tv @ ( F @ X3 ) @ B ) )
=> ( ord_le241064091lle_tv @ ( image_1151963140lle_tv @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_312_image__subsetI,axiom,
! [A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > nat,B: set_nat] :
( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_376325286tv_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_313_image__subsetI,axiom,
! [A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv,B: set_Pa1764573435lle_tv] :
( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ A )
=> ( member266900804lle_tv @ ( F @ X3 ) @ B ) )
=> ( ord_le241064091lle_tv @ ( image_335818702lle_tv @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_314_top_Oextremum__uniqueI,axiom,
! [A4: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ top_to1057771083lle_tv @ A4 )
=> ( A4 = top_to1057771083lle_tv ) ) ).
% top.extremum_uniqueI
thf(fact_315_top_Oextremum__uniqueI,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
=> ( A4 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_316_top_Oextremum__unique,axiom,
! [A4: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ top_to1057771083lle_tv @ A4 )
= ( A4 = top_to1057771083lle_tv ) ) ).
% top.extremum_unique
thf(fact_317_top_Oextremum__unique,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
= ( A4 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_318_top__greatest,axiom,
! [A4: set_Pa1764573435lle_tv] : ( ord_le241064091lle_tv @ A4 @ top_to1057771083lle_tv ) ).
% top_greatest
thf(fact_319_top__greatest,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).
% top_greatest
thf(fact_320_subset__UNIV,axiom,
! [A: set_Pa1764573435lle_tv] : ( ord_le241064091lle_tv @ A @ top_to1057771083lle_tv ) ).
% subset_UNIV
thf(fact_321_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_322_subset__inj__on,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,B: set_Pa1764573435lle_tv,A: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F @ B )
=> ( ( ord_le241064091lle_tv @ A @ B )
=> ( inj_on772319074lle_tv @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_323_subset__inj__on,axiom,
! [F: nat > nat,B: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_324_inj__on__subset,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F @ A )
=> ( ( ord_le241064091lle_tv @ B @ A )
=> ( inj_on772319074lle_tv @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_325_inj__on__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( inj_on_nat_nat @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_326_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_327_in__mono,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv] :
( ( ord_le241064091lle_tv @ A @ B )
=> ( ( member266900804lle_tv @ X @ A )
=> ( member266900804lle_tv @ X @ B ) ) ) ).
% in_mono
thf(fact_328_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_329_subsetD,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv,C: paraco1605129243lle_tv] :
( ( ord_le241064091lle_tv @ A @ B )
=> ( ( member266900804lle_tv @ C @ A )
=> ( member266900804lle_tv @ C @ B ) ) ) ).
% subsetD
thf(fact_330_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_331_subset__eq,axiom,
( ord_le241064091lle_tv
= ( ^ [A2: set_Pa1764573435lle_tv,B3: set_Pa1764573435lle_tv] :
! [X2: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X2 @ A2 )
=> ( member266900804lle_tv @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_332_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A2 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_333_subset__iff,axiom,
( ord_le241064091lle_tv
= ( ^ [A2: set_Pa1764573435lle_tv,B3: set_Pa1764573435lle_tv] :
! [T: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ T @ A2 )
=> ( member266900804lle_tv @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_334_linorder__inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) ) ) )
=> ( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% linorder_inj_onI
thf(fact_335_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B: set_nat,A4: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( member_nat @ A4 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_336_inj__on__image__mem__iff,axiom,
! [F: nat > paraco1605129243lle_tv,B: set_nat,A4: nat,A: set_nat] :
( ( inj_on1955349912lle_tv @ F @ B )
=> ( ( member_nat @ A4 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member266900804lle_tv @ ( F @ A4 ) @ ( image_1151963140lle_tv @ F @ A ) )
= ( member_nat @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_337_inj__on__image__mem__iff,axiom,
! [F: paraco1605129243lle_tv > nat,B: set_Pa1764573435lle_tv,A4: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( inj_on1179712058tv_nat @ F @ B )
=> ( ( member266900804lle_tv @ A4 @ B )
=> ( ( ord_le241064091lle_tv @ A @ B )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_376325286tv_nat @ F @ A ) )
= ( member266900804lle_tv @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_338_inj__on__image__mem__iff,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,B: set_Pa1764573435lle_tv,A4: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F @ B )
=> ( ( member266900804lle_tv @ A4 @ B )
=> ( ( ord_le241064091lle_tv @ A @ B )
=> ( ( member266900804lle_tv @ ( F @ A4 ) @ ( image_335818702lle_tv @ F @ A ) )
= ( member266900804lle_tv @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_339_inj__on__image__eq__iff,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,C2: set_Pa1764573435lle_tv,A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F @ C2 )
=> ( ( ord_le241064091lle_tv @ A @ C2 )
=> ( ( ord_le241064091lle_tv @ B @ C2 )
=> ( ( ( image_335818702lle_tv @ F @ A )
= ( image_335818702lle_tv @ F @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_340_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C2: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_341_inj__image__subset__iff,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F @ top_to1057771083lle_tv )
=> ( ( ord_le241064091lle_tv @ ( image_335818702lle_tv @ F @ A ) @ ( image_335818702lle_tv @ F @ B ) )
= ( ord_le241064091lle_tv @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_342_inj__image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_343_map__prod__surj,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,G: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( ( image_335818702lle_tv @ F @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ( ( ( image_335818702lle_tv @ G @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ( ( image_1221801032lle_tv @ ( produc1345917071lle_tv @ F @ G ) @ top_to1871154984lle_tv )
= top_to1871154984lle_tv ) ) ) ).
% map_prod_surj
thf(fact_344_map__prod__surj,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,G: paraco1605129243lle_tv > nat] :
( ( ( image_335818702lle_tv @ F @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ( ( ( image_376325286tv_nat @ G @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ( ( image_1796815776tv_nat @ ( produc1500159719tv_nat @ F @ G ) @ top_to1871154984lle_tv )
= top_to975729152tv_nat ) ) ) ).
% map_prod_surj
thf(fact_345_map__prod__surj,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,G: nat > paraco1605129243lle_tv] :
( ( ( image_335818702lle_tv @ F @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ( ( ( image_1151963140lle_tv @ G @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ( ( image_228488944lle_tv @ ( produc1521089895lle_tv @ F @ G ) @ top_to975729152tv_nat )
= top_to1871154984lle_tv ) ) ) ).
% map_prod_surj
thf(fact_346_map__prod__surj,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,G: nat > nat] :
( ( ( image_335818702lle_tv @ F @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_1104296520tv_nat @ ( produc42536895at_nat @ F @ G ) @ top_to975729152tv_nat )
= top_to975729152tv_nat ) ) ) ).
% map_prod_surj
thf(fact_347_map__prod__surj,axiom,
! [F: paraco1605129243lle_tv > nat,G: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( ( image_376325286tv_nat @ F @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ( ( ( image_335818702lle_tv @ G @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ( ( image_1892891680lle_tv @ ( produc1095080935lle_tv @ F @ G ) @ top_to1871154984lle_tv )
= top_to1411004326lle_tv ) ) ) ).
% map_prod_surj
thf(fact_348_map__prod__surj,axiom,
! [F: paraco1605129243lle_tv > nat,G: paraco1605129243lle_tv > nat] :
( ( ( image_376325286tv_nat @ F @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ( ( ( image_376325286tv_nat @ G @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ( ( image_901203832at_nat @ ( produc51277375tv_nat @ F @ G ) @ top_to1871154984lle_tv )
= top_to1555900337at_nat ) ) ) ).
% map_prod_surj
thf(fact_349_map__prod__surj,axiom,
! [F: paraco1605129243lle_tv > nat,G: nat > paraco1605129243lle_tv] :
( ( ( image_376325286tv_nat @ F @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ( ( ( image_1151963140lle_tv @ G @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ( ( image_1200372424lle_tv @ ( produc72207551lle_tv @ F @ G ) @ top_to975729152tv_nat )
= top_to1411004326lle_tv ) ) ) ).
% map_prod_surj
thf(fact_350_map__prod__surj,axiom,
! [F: paraco1605129243lle_tv > nat,G: nat > nat] :
( ( ( image_376325286tv_nat @ F @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_1805124128at_nat @ ( produc386540311at_nat @ F @ G ) @ top_to975729152tv_nat )
= top_to1555900337at_nat ) ) ) ).
% map_prod_surj
thf(fact_351_map__prod__surj,axiom,
! [F: nat > paraco1605129243lle_tv,G: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( ( image_1151963140lle_tv @ F @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ( ( ( image_335818702lle_tv @ G @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ( ( image_988201162lle_tv @ ( produc1975500101lle_tv @ F @ G ) @ top_to1411004326lle_tv )
= top_to1871154984lle_tv ) ) ) ).
% map_prod_surj
thf(fact_352_map__prod__surj,axiom,
! [F: nat > paraco1605129243lle_tv,G: paraco1605129243lle_tv > nat] :
( ( ( image_1151963140lle_tv @ F @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ( ( ( image_376325286tv_nat @ G @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ( ( image_53754658tv_nat @ ( produc189459613tv_nat @ F @ G ) @ top_to1411004326lle_tv )
= top_to975729152tv_nat ) ) ) ).
% map_prod_surj
thf(fact_353_subset__image__inj,axiom,
! [S4: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv,T2: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ S4 @ ( image_335818702lle_tv @ F @ T2 ) )
= ( ? [U: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ U @ T2 )
& ( inj_on772319074lle_tv @ F @ U )
& ( S4
= ( image_335818702lle_tv @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_354_subset__image__inj,axiom,
! [S4: set_nat,F: nat > nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S4 @ ( image_nat_nat @ F @ T2 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T2 )
& ( inj_on_nat_nat @ F @ U )
& ( S4
= ( image_nat_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
% Conjectures (1)
thf(conj_0,conjecture,
inj_on772319074lle_tv @ ( paraco1147068288nge_tv @ f ) @ top_to1057771083lle_tv ).
%------------------------------------------------------------------------------