TPTP Problem File: ITP098^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP098^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer ListInf problem prob_167__5410008_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 : ListInf/prob_167__5410008_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 434 ( 110 unt; 78 typ; 0 def)
% Number of atoms : 1026 ( 619 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 3023 ( 32 ~; 0 |; 8 &;2519 @)
% ( 0 <=>; 464 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 857 ( 857 >; 0 *; 0 +; 0 <<)
% Number of symbols : 69 ( 68 usr; 7 con; 0-3 aty)
% Number of variables : 1155 ( 49 ^;1070 !; 36 ?;1155 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:31:01.240
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
set_c_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
set_a_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
set_a_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__c_J,type,
set_c: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__c,type,
c: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (68)
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__c,type,
comp_a_a_c: ( a > a ) > ( c > a ) > c > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__a,type,
comp_a_b_a: ( a > b ) > ( a > a ) > a > b ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__b,type,
comp_a_b_b: ( a > b ) > ( b > a ) > b > b ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__c,type,
comp_a_b_c: ( a > b ) > ( c > a ) > c > b ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__a,type,
comp_a_c_a: ( a > c ) > ( a > a ) > a > c ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__c,type,
comp_a_c_c: ( a > c ) > ( c > a ) > c > c ).
thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__a,type,
comp_b_a_a: ( b > a ) > ( a > b ) > a > a ).
thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__c,type,
comp_b_a_c: ( b > a ) > ( c > b ) > c > a ).
thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__a,type,
comp_b_b_a: ( b > b ) > ( a > b ) > a > b ).
thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__b,type,
comp_b_b_b: ( b > b ) > ( b > b ) > b > b ).
thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__c,type,
comp_b_b_c: ( b > b ) > ( c > b ) > c > b ).
thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001tf__a,type,
comp_b_c_a: ( b > c ) > ( a > b ) > a > c ).
thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001tf__c,type,
comp_b_c_c: ( b > c ) > ( c > b ) > c > c ).
thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__a,type,
comp_c_a_a: ( c > a ) > ( a > c ) > a > a ).
thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__b,type,
comp_c_a_b: ( c > a ) > ( b > c ) > b > a ).
thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__c,type,
comp_c_a_c: ( c > a ) > ( c > c ) > c > a ).
thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__a,type,
comp_c_b_a: ( c > b ) > ( a > c ) > a > b ).
thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__b,type,
comp_c_b_b: ( c > b ) > ( b > c ) > b > b ).
thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__c,type,
comp_c_b_c: ( c > b ) > ( c > c ) > c > b ).
thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001tf__a,type,
comp_c_c_a: ( c > c ) > ( a > c ) > a > c ).
thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001tf__c,type,
comp_c_c_c: ( c > c ) > ( c > c ) > c > c ).
thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__b_J,type,
inj_on_a_a_a_b: ( ( a > a ) > a > b ) > set_a_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mtf__b_J_001_062_Itf__a_Mtf__b_J,type,
inj_on_a_b_a_b: ( ( a > b ) > a > b ) > set_a_b > $o ).
thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mtf__c_J_001_062_Itf__a_Mtf__b_J,type,
inj_on_a_c_a_b: ( ( a > c ) > a > b ) > set_a_c > $o ).
thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
inj_on_c_a_c_a: ( ( c > a ) > c > a ) > set_c_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__b_J,type,
inj_on_c_a_c_b: ( ( c > a ) > c > b ) > set_c_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__c_J,type,
inj_on_c_a_c_c: ( ( c > a ) > c > c ) > set_c_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001tf__a,type,
inj_on_c_a_a: ( ( c > a ) > a ) > set_c_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__b,type,
inj_on_a_b: ( a > b ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__c,type,
inj_on_a_c: ( a > c ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001tf__a,type,
inj_on_b_a: ( b > a ) > set_b > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001tf__b,type,
inj_on_b_b: ( b > b ) > set_b > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001tf__c,type,
inj_on_b_c: ( b > c ) > set_b > $o ).
thf(sy_c_Fun_Oinj__on_001tf__c_001tf__a,type,
inj_on_c_a: ( c > a ) > set_c > $o ).
thf(sy_c_Fun_Oinj__on_001tf__c_001tf__b,type,
inj_on_c_b: ( c > b ) > set_c > $o ).
thf(sy_c_Fun_Oinj__on_001tf__c_001tf__c,type,
inj_on_c_c: ( c > c ) > set_c > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
sup_sup_set_c_a: set_c_a > set_c_a > set_c_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
sup_sup_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__c_J,type,
sup_sup_set_c: set_c > set_c > set_c ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
top_top_set_a_a: set_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
top_top_set_a_b: set_a_b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
top_top_set_a_c: set_a_c ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
top_top_set_c_a: set_c_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__b_J,type,
top_top_set_b: set_b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__c_J,type,
top_top_set_c: set_c ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_062_Itf__c_Mtf__a_J_001tf__a,type,
image_c_a_a: ( ( c > a ) > a ) > set_c_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__b,type,
image_a_b: ( a > b ) > set_a > set_b ).
thf(sy_c_Set_Oimage_001tf__a_001tf__c,type,
image_a_c: ( a > c ) > set_a > set_c ).
thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
image_b_a: ( b > a ) > set_b > set_a ).
thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Oimage_001tf__b_001tf__c,type,
image_b_c: ( b > c ) > set_b > set_c ).
thf(sy_c_Set_Oimage_001tf__c_001_062_Itf__c_Mtf__a_J,type,
image_c_c_a: ( c > c > a ) > set_c > set_c_a ).
thf(sy_c_Set_Oimage_001tf__c_001tf__a,type,
image_c_a: ( c > a ) > set_c > set_a ).
thf(sy_c_Set_Oimage_001tf__c_001tf__b,type,
image_c_b: ( c > b ) > set_c > set_b ).
thf(sy_c_Set_Oimage_001tf__c_001tf__c,type,
image_c_c: ( c > c ) > set_c > set_c ).
thf(sy_c_member_001_062_Itf__c_Mtf__a_J,type,
member_c_a: ( c > a ) > set_c_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_c_member_001tf__c,type,
member_c: c > set_c > $o ).
thf(sy_v_f,type,
f: a > b ).
thf(sy_v_g,type,
g: c > a ).
thf(sy_v_h,type,
h: c > a ).
% Relevant facts (353)
thf(fact_0_o__ext,axiom,
! [H: a > c,F: c > b,G: c > b] :
( ! [X: c] :
( ( member_c @ X @ ( image_a_c @ H @ top_top_set_a ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( comp_c_b_a @ F @ H )
= ( comp_c_b_a @ G @ H ) ) ) ).
% o_ext
thf(fact_1_o__ext,axiom,
! [H: a > b,F: b > b,G: b > b] :
( ! [X: b] :
( ( member_b @ X @ ( image_a_b @ H @ top_top_set_a ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( comp_b_b_a @ F @ H )
= ( comp_b_b_a @ G @ H ) ) ) ).
% o_ext
thf(fact_2_o__ext,axiom,
! [H: a > a,F: a > b,G: a > b] :
( ! [X: a] :
( ( member_a @ X @ ( image_a_a @ H @ top_top_set_a ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( comp_a_b_a @ F @ H )
= ( comp_a_b_a @ G @ H ) ) ) ).
% o_ext
thf(fact_3_o__ext,axiom,
! [H: c > a,F: a > b,G: a > b] :
( ! [X: a] :
( ( member_a @ X @ ( image_c_a @ H @ top_top_set_c ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( comp_a_b_c @ F @ H )
= ( comp_a_b_c @ G @ H ) ) ) ).
% o_ext
thf(fact_4_o__cong,axiom,
! [H: a > c,I: a > c,F: c > b,G: c > b] :
( ( H = I )
=> ( ! [X: c] :
( ( member_c @ X @ ( image_a_c @ I @ top_top_set_a ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( comp_c_b_a @ F @ H )
= ( comp_c_b_a @ F @ I ) ) ) ) ).
% o_cong
thf(fact_5_o__cong,axiom,
! [H: a > b,I: a > b,F: b > b,G: b > b] :
( ( H = I )
=> ( ! [X: b] :
( ( member_b @ X @ ( image_a_b @ I @ top_top_set_a ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( comp_b_b_a @ F @ H )
= ( comp_b_b_a @ F @ I ) ) ) ) ).
% o_cong
thf(fact_6_o__cong,axiom,
! [H: a > a,I: a > a,F: a > b,G: a > b] :
( ( H = I )
=> ( ! [X: a] :
( ( member_a @ X @ ( image_a_a @ I @ top_top_set_a ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( comp_a_b_a @ F @ H )
= ( comp_a_b_a @ F @ I ) ) ) ) ).
% o_cong
thf(fact_7_o__cong,axiom,
! [H: c > a,I: c > a,F: a > b,G: a > b] :
( ( H = I )
=> ( ! [X: a] :
( ( member_a @ X @ ( image_c_a @ I @ top_top_set_c ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( comp_a_b_c @ F @ H )
= ( comp_a_b_c @ F @ I ) ) ) ) ).
% o_cong
thf(fact_8_o__inj__on,axiom,
! [F: a > b,G: c > a,H: c > a] :
( ( ( comp_a_b_c @ F @ G )
= ( comp_a_b_c @ F @ H ) )
=> ( ( inj_on_a_b @ F @ ( sup_sup_set_a @ ( image_c_a @ G @ top_top_set_c ) @ ( image_c_a @ H @ top_top_set_c ) ) )
=> ( G = H ) ) ) ).
% o_inj_on
thf(fact_9_o__inj__on,axiom,
! [F: b > b,G: c > b,H: c > b] :
( ( ( comp_b_b_c @ F @ G )
= ( comp_b_b_c @ F @ H ) )
=> ( ( inj_on_b_b @ F @ ( sup_sup_set_b @ ( image_c_b @ G @ top_top_set_c ) @ ( image_c_b @ H @ top_top_set_c ) ) )
=> ( G = H ) ) ) ).
% o_inj_on
thf(fact_10_o__inj__on,axiom,
! [F: a > c,G: c > a,H: c > a] :
( ( ( comp_a_c_c @ F @ G )
= ( comp_a_c_c @ F @ H ) )
=> ( ( inj_on_a_c @ F @ ( sup_sup_set_a @ ( image_c_a @ G @ top_top_set_c ) @ ( image_c_a @ H @ top_top_set_c ) ) )
=> ( G = H ) ) ) ).
% o_inj_on
thf(fact_11_o__inj__on,axiom,
! [F: a > a,G: c > a,H: c > a] :
( ( ( comp_a_a_c @ F @ G )
= ( comp_a_a_c @ F @ H ) )
=> ( ( inj_on_a_a @ F @ ( sup_sup_set_a @ ( image_c_a @ G @ top_top_set_c ) @ ( image_c_a @ H @ top_top_set_c ) ) )
=> ( G = H ) ) ) ).
% o_inj_on
thf(fact_12_o__inj__on,axiom,
! [F: c > b,G: c > c,H: c > c] :
( ( ( comp_c_b_c @ F @ G )
= ( comp_c_b_c @ F @ H ) )
=> ( ( inj_on_c_b @ F @ ( sup_sup_set_c @ ( image_c_c @ G @ top_top_set_c ) @ ( image_c_c @ H @ top_top_set_c ) ) )
=> ( G = H ) ) ) ).
% o_inj_on
thf(fact_13_o__inj__on,axiom,
! [F: c > a,G: c > c,H: c > c] :
( ( ( comp_c_a_c @ F @ G )
= ( comp_c_a_c @ F @ H ) )
=> ( ( inj_on_c_a @ F @ ( sup_sup_set_c @ ( image_c_c @ G @ top_top_set_c ) @ ( image_c_c @ H @ top_top_set_c ) ) )
=> ( G = H ) ) ) ).
% o_inj_on
thf(fact_14_o__inj__on,axiom,
! [F: b > b,G: a > b,H: a > b] :
( ( ( comp_b_b_a @ F @ G )
= ( comp_b_b_a @ F @ H ) )
=> ( ( inj_on_b_b @ F @ ( sup_sup_set_b @ ( image_a_b @ G @ top_top_set_a ) @ ( image_a_b @ H @ top_top_set_a ) ) )
=> ( G = H ) ) ) ).
% o_inj_on
thf(fact_15_o__inj__on,axiom,
! [F: a > b,G: a > a,H: a > a] :
( ( ( comp_a_b_a @ F @ G )
= ( comp_a_b_a @ F @ H ) )
=> ( ( inj_on_a_b @ F @ ( sup_sup_set_a @ ( image_a_a @ G @ top_top_set_a ) @ ( image_a_a @ H @ top_top_set_a ) ) )
=> ( G = H ) ) ) ).
% o_inj_on
thf(fact_16_o__inj__on,axiom,
! [F: a > c,G: a > a,H: a > a] :
( ( ( comp_a_c_a @ F @ G )
= ( comp_a_c_a @ F @ H ) )
=> ( ( inj_on_a_c @ F @ ( sup_sup_set_a @ ( image_a_a @ G @ top_top_set_a ) @ ( image_a_a @ H @ top_top_set_a ) ) )
=> ( G = H ) ) ) ).
% o_inj_on
thf(fact_17_o__inj__on,axiom,
! [F: a > a,G: a > a,H: a > a] :
( ( ( comp_a_a_a @ F @ G )
= ( comp_a_a_a @ F @ H ) )
=> ( ( inj_on_a_a @ F @ ( sup_sup_set_a @ ( image_a_a @ G @ top_top_set_a ) @ ( image_a_a @ H @ top_top_set_a ) ) )
=> ( G = H ) ) ) ).
% o_inj_on
thf(fact_18_ex__o__conv,axiom,
! [G: a > b,F: c > b] :
( ( ? [H2: a > c] :
( G
= ( comp_c_b_a @ F @ H2 ) ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ ( image_a_b @ G @ top_top_set_a ) )
=> ? [Y: c] :
( X2
= ( F @ Y ) ) ) ) ) ).
% ex_o_conv
thf(fact_19_ex__o__conv,axiom,
! [G: a > b,F: a > b] :
( ( ? [H2: a > a] :
( G
= ( comp_a_b_a @ F @ H2 ) ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ ( image_a_b @ G @ top_top_set_a ) )
=> ? [Y: a] :
( X2
= ( F @ Y ) ) ) ) ) ).
% ex_o_conv
thf(fact_20_ex__o__conv,axiom,
! [G: a > b,F: b > b] :
( ( ? [H2: a > b] :
( G
= ( comp_b_b_a @ F @ H2 ) ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ ( image_a_b @ G @ top_top_set_a ) )
=> ? [Y: b] :
( X2
= ( F @ Y ) ) ) ) ) ).
% ex_o_conv
thf(fact_21_ex__o__conv,axiom,
! [G: c > b,F: a > b] :
( ( ? [H2: c > a] :
( G
= ( comp_a_b_c @ F @ H2 ) ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ ( image_c_b @ G @ top_top_set_c ) )
=> ? [Y: a] :
( X2
= ( F @ Y ) ) ) ) ) ).
% ex_o_conv
thf(fact_22_o__eq__conv,axiom,
! [F: c > b,H: a > c,G: c > b] :
( ( ( comp_c_b_a @ F @ H )
= ( comp_c_b_a @ G @ H ) )
= ( ! [X2: c] :
( ( member_c @ X2 @ ( image_a_c @ H @ top_top_set_a ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% o_eq_conv
thf(fact_23_o__eq__conv,axiom,
! [F: a > b,H: a > a,G: a > b] :
( ( ( comp_a_b_a @ F @ H )
= ( comp_a_b_a @ G @ H ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ H @ top_top_set_a ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% o_eq_conv
thf(fact_24_o__eq__conv,axiom,
! [F: b > b,H: a > b,G: b > b] :
( ( ( comp_b_b_a @ F @ H )
= ( comp_b_b_a @ G @ H ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ ( image_a_b @ H @ top_top_set_a ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% o_eq_conv
thf(fact_25_o__eq__conv,axiom,
! [F: a > b,H: c > a,G: a > b] :
( ( ( comp_a_b_c @ F @ H )
= ( comp_a_b_c @ G @ H ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( image_c_a @ H @ top_top_set_c ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% o_eq_conv
thf(fact_26_sup__top__left,axiom,
! [X3: set_b] :
( ( sup_sup_set_b @ top_top_set_b @ X3 )
= top_top_set_b ) ).
% sup_top_left
thf(fact_27_sup__top__left,axiom,
! [X3: set_c_a] :
( ( sup_sup_set_c_a @ top_top_set_c_a @ X3 )
= top_top_set_c_a ) ).
% sup_top_left
thf(fact_28_sup__top__left,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X3 )
= top_top_set_a ) ).
% sup_top_left
thf(fact_29_sup__top__left,axiom,
! [X3: set_c] :
( ( sup_sup_set_c @ top_top_set_c @ X3 )
= top_top_set_c ) ).
% sup_top_left
thf(fact_30_sup__top__right,axiom,
! [X3: set_b] :
( ( sup_sup_set_b @ X3 @ top_top_set_b )
= top_top_set_b ) ).
% sup_top_right
thf(fact_31_sup__top__right,axiom,
! [X3: set_c_a] :
( ( sup_sup_set_c_a @ X3 @ top_top_set_c_a )
= top_top_set_c_a ) ).
% sup_top_right
thf(fact_32_sup__top__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ top_top_set_a )
= top_top_set_a ) ).
% sup_top_right
thf(fact_33_sup__top__right,axiom,
! [X3: set_c] :
( ( sup_sup_set_c @ X3 @ top_top_set_c )
= top_top_set_c ) ).
% sup_top_right
thf(fact_34_inj__on__Un__image__eq__iff,axiom,
! [F: b > b,A: set_b,B: set_b] :
( ( inj_on_b_b @ F @ ( sup_sup_set_b @ A @ B ) )
=> ( ( ( image_b_b @ F @ A )
= ( image_b_b @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_35_inj__on__Un__image__eq__iff,axiom,
! [F: a > c,A: set_a,B: set_a] :
( ( inj_on_a_c @ F @ ( sup_sup_set_a @ A @ B ) )
=> ( ( ( image_a_c @ F @ A )
= ( image_a_c @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_36_inj__on__Un__image__eq__iff,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( inj_on_a_a @ F @ ( sup_sup_set_a @ A @ B ) )
=> ( ( ( image_a_a @ F @ A )
= ( image_a_a @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_37_inj__on__Un__image__eq__iff,axiom,
! [F: c > b,A: set_c,B: set_c] :
( ( inj_on_c_b @ F @ ( sup_sup_set_c @ A @ B ) )
=> ( ( ( image_c_b @ F @ A )
= ( image_c_b @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_38_inj__on__Un__image__eq__iff,axiom,
! [F: c > a,A: set_c,B: set_c] :
( ( inj_on_c_a @ F @ ( sup_sup_set_c @ A @ B ) )
=> ( ( ( image_c_a @ F @ A )
= ( image_c_a @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_39_inj__on__Un__image__eq__iff,axiom,
! [F: a > b,A: set_a,B: set_a] :
( ( inj_on_a_b @ F @ ( sup_sup_set_a @ A @ B ) )
=> ( ( ( image_a_b @ F @ A )
= ( image_a_b @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_40_inj__compose,axiom,
! [F: b > b,G: a > b] :
( ( inj_on_b_b @ F @ top_top_set_b )
=> ( ( inj_on_a_b @ G @ top_top_set_a )
=> ( inj_on_a_b @ ( comp_b_b_a @ F @ G ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_41_inj__compose,axiom,
! [F: a > b,G: a > a] :
( ( inj_on_a_b @ F @ top_top_set_a )
=> ( ( inj_on_a_a @ G @ top_top_set_a )
=> ( inj_on_a_b @ ( comp_a_b_a @ F @ G ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_42_inj__compose,axiom,
! [F: a > b,G: c > a] :
( ( inj_on_a_b @ F @ top_top_set_a )
=> ( ( inj_on_c_a @ G @ top_top_set_c )
=> ( inj_on_c_b @ ( comp_a_b_c @ F @ G ) @ top_top_set_c ) ) ) ).
% inj_compose
thf(fact_43_inj__compose,axiom,
! [F: c > b,G: a > c] :
( ( inj_on_c_b @ F @ top_top_set_c )
=> ( ( inj_on_a_c @ G @ top_top_set_a )
=> ( inj_on_a_b @ ( comp_c_b_a @ F @ G ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_44_inj__compose,axiom,
! [F: c > b,G: c > c] :
( ( inj_on_c_b @ F @ top_top_set_c )
=> ( ( inj_on_c_c @ G @ top_top_set_c )
=> ( inj_on_c_b @ ( comp_c_b_c @ F @ G ) @ top_top_set_c ) ) ) ).
% inj_compose
thf(fact_45_inj__compose,axiom,
! [F: c > a,G: c > c] :
( ( inj_on_c_a @ F @ top_top_set_c )
=> ( ( inj_on_c_c @ G @ top_top_set_c )
=> ( inj_on_c_a @ ( comp_c_a_c @ F @ G ) @ top_top_set_c ) ) ) ).
% inj_compose
thf(fact_46_inj__compose,axiom,
! [F: c > c,G: a > c] :
( ( inj_on_c_c @ F @ top_top_set_c )
=> ( ( inj_on_a_c @ G @ top_top_set_a )
=> ( inj_on_a_c @ ( comp_c_c_a @ F @ G ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_47_inj__compose,axiom,
! [F: c > a,G: a > c] :
( ( inj_on_c_a @ F @ top_top_set_c )
=> ( ( inj_on_a_c @ G @ top_top_set_a )
=> ( inj_on_a_a @ ( comp_c_a_a @ F @ G ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_48_inj__compose,axiom,
! [F: c > b,G: b > c] :
( ( inj_on_c_b @ F @ top_top_set_c )
=> ( ( inj_on_b_c @ G @ top_top_set_b )
=> ( inj_on_b_b @ ( comp_c_b_b @ F @ G ) @ top_top_set_b ) ) ) ).
% inj_compose
thf(fact_49_inj__compose,axiom,
! [F: c > a,G: b > c] :
( ( inj_on_c_a @ F @ top_top_set_c )
=> ( ( inj_on_b_c @ G @ top_top_set_b )
=> ( inj_on_b_a @ ( comp_c_a_b @ F @ G ) @ top_top_set_b ) ) ) ).
% inj_compose
thf(fact_50_fun_Oinj__map,axiom,
! [F: c > b] :
( ( inj_on_c_b @ F @ top_top_set_c )
=> ( inj_on_a_c_a_b @ ( comp_c_b_a @ F ) @ top_top_set_a_c ) ) ).
% fun.inj_map
thf(fact_51_fun_Oinj__map,axiom,
! [F: a > b] :
( ( inj_on_a_b @ F @ top_top_set_a )
=> ( inj_on_a_a_a_b @ ( comp_a_b_a @ F ) @ top_top_set_a_a ) ) ).
% fun.inj_map
thf(fact_52_fun_Oinj__map,axiom,
! [F: a > c] :
( ( inj_on_a_c @ F @ top_top_set_a )
=> ( inj_on_c_a_c_c @ ( comp_a_c_c @ F ) @ top_top_set_c_a ) ) ).
% fun.inj_map
thf(fact_53_fun_Oinj__map,axiom,
! [F: a > a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( inj_on_c_a_c_a @ ( comp_a_a_c @ F ) @ top_top_set_c_a ) ) ).
% fun.inj_map
thf(fact_54_fun_Oinj__map,axiom,
! [F: b > b] :
( ( inj_on_b_b @ F @ top_top_set_b )
=> ( inj_on_a_b_a_b @ ( comp_b_b_a @ F ) @ top_top_set_a_b ) ) ).
% fun.inj_map
thf(fact_55_fun_Oinj__map,axiom,
! [F: a > b] :
( ( inj_on_a_b @ F @ top_top_set_a )
=> ( inj_on_c_a_c_b @ ( comp_a_b_c @ F ) @ top_top_set_c_a ) ) ).
% fun.inj_map
thf(fact_56_comp__inj__on,axiom,
! [F: b > a,A: set_b,G: a > b] :
( ( inj_on_b_a @ F @ A )
=> ( ( inj_on_a_b @ G @ ( image_b_a @ F @ A ) )
=> ( inj_on_b_b @ ( comp_a_b_b @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_57_comp__inj__on,axiom,
! [F: c > c,A: set_c,G: c > b] :
( ( inj_on_c_c @ F @ A )
=> ( ( inj_on_c_b @ G @ ( image_c_c @ F @ A ) )
=> ( inj_on_c_b @ ( comp_c_b_c @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_58_comp__inj__on,axiom,
! [F: b > c,A: set_b,G: c > b] :
( ( inj_on_b_c @ F @ A )
=> ( ( inj_on_c_b @ G @ ( image_b_c @ F @ A ) )
=> ( inj_on_b_b @ ( comp_c_b_b @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_59_comp__inj__on,axiom,
! [F: c > c,A: set_c,G: c > a] :
( ( inj_on_c_c @ F @ A )
=> ( ( inj_on_c_a @ G @ ( image_c_c @ F @ A ) )
=> ( inj_on_c_a @ ( comp_c_a_c @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_60_comp__inj__on,axiom,
! [F: a > b,A: set_a,G: b > c] :
( ( inj_on_a_b @ F @ A )
=> ( ( inj_on_b_c @ G @ ( image_a_b @ F @ A ) )
=> ( inj_on_a_c @ ( comp_b_c_a @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_61_comp__inj__on,axiom,
! [F: a > b,A: set_a,G: b > a] :
( ( inj_on_a_b @ F @ A )
=> ( ( inj_on_b_a @ G @ ( image_a_b @ F @ A ) )
=> ( inj_on_a_a @ ( comp_b_a_a @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_62_comp__inj__on,axiom,
! [F: a > b,A: set_a,G: b > b] :
( ( inj_on_a_b @ F @ A )
=> ( ( inj_on_b_b @ G @ ( image_a_b @ F @ A ) )
=> ( inj_on_a_b @ ( comp_b_b_a @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_63_comp__inj__on,axiom,
! [F: c > b,A: set_c,G: b > a] :
( ( inj_on_c_b @ F @ A )
=> ( ( inj_on_b_a @ G @ ( image_c_b @ F @ A ) )
=> ( inj_on_c_a @ ( comp_b_a_c @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_64_comp__inj__on,axiom,
! [F: c > b,A: set_c,G: b > b] :
( ( inj_on_c_b @ F @ A )
=> ( ( inj_on_b_b @ G @ ( image_c_b @ F @ A ) )
=> ( inj_on_c_b @ ( comp_b_b_c @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_65_comp__inj__on,axiom,
! [F: c > a,A: set_c,G: a > b] :
( ( inj_on_c_a @ F @ A )
=> ( ( inj_on_a_b @ G @ ( image_c_a @ F @ A ) )
=> ( inj_on_c_b @ ( comp_a_b_c @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_66_inj__on__imageI,axiom,
! [G: a > c,F: c > a,A: set_c] :
( ( inj_on_c_c @ ( comp_a_c_c @ G @ F ) @ A )
=> ( inj_on_a_c @ G @ ( image_c_a @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_67_inj__on__imageI,axiom,
! [G: a > b,F: a > a,A: set_a] :
( ( inj_on_a_b @ ( comp_a_b_a @ G @ F ) @ A )
=> ( inj_on_a_b @ G @ ( image_a_a @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_68_inj__on__imageI,axiom,
! [G: c > b,F: a > c,A: set_a] :
( ( inj_on_a_b @ ( comp_c_b_a @ G @ F ) @ A )
=> ( inj_on_c_b @ G @ ( image_a_c @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_69_inj__on__imageI,axiom,
! [G: b > b,F: a > b,A: set_a] :
( ( inj_on_a_b @ ( comp_b_b_a @ G @ F ) @ A )
=> ( inj_on_b_b @ G @ ( image_a_b @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_70_inj__on__imageI,axiom,
! [G: a > b,F: c > a,A: set_c] :
( ( inj_on_c_b @ ( comp_a_b_c @ G @ F ) @ A )
=> ( inj_on_a_b @ G @ ( image_c_a @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_71_inj__on__imageI,axiom,
! [G: c > b,F: c > c,A: set_c] :
( ( inj_on_c_b @ ( comp_c_b_c @ G @ F ) @ A )
=> ( inj_on_c_b @ G @ ( image_c_c @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_72_inj__on__imageI,axiom,
! [G: b > b,F: c > b,A: set_c] :
( ( inj_on_c_b @ ( comp_b_b_c @ G @ F ) @ A )
=> ( inj_on_b_b @ G @ ( image_c_b @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_73_inj__on__imageI,axiom,
! [G: b > a,F: c > b,A: set_c] :
( ( inj_on_c_a @ ( comp_b_a_c @ G @ F ) @ A )
=> ( inj_on_b_a @ G @ ( image_c_b @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_74_inj__on__imageI,axiom,
! [G: c > a,F: c > c,A: set_c] :
( ( inj_on_c_a @ ( comp_c_a_c @ G @ F ) @ A )
=> ( inj_on_c_a @ G @ ( image_c_c @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_75_inj__on__imageI,axiom,
! [G: a > a,F: c > a,A: set_c] :
( ( inj_on_c_a @ ( comp_a_a_c @ G @ F ) @ A )
=> ( inj_on_a_a @ G @ ( image_c_a @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_76_comp__inj__on__iff,axiom,
! [F: b > a,A: set_b,F2: a > b] :
( ( inj_on_b_a @ F @ A )
=> ( ( inj_on_a_b @ F2 @ ( image_b_a @ F @ A ) )
= ( inj_on_b_b @ ( comp_a_b_b @ F2 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_77_comp__inj__on__iff,axiom,
! [F: c > c,A: set_c,F2: c > b] :
( ( inj_on_c_c @ F @ A )
=> ( ( inj_on_c_b @ F2 @ ( image_c_c @ F @ A ) )
= ( inj_on_c_b @ ( comp_c_b_c @ F2 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_78_comp__inj__on__iff,axiom,
! [F: b > c,A: set_b,F2: c > b] :
( ( inj_on_b_c @ F @ A )
=> ( ( inj_on_c_b @ F2 @ ( image_b_c @ F @ A ) )
= ( inj_on_b_b @ ( comp_c_b_b @ F2 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_79_comp__inj__on__iff,axiom,
! [F: c > c,A: set_c,F2: c > a] :
( ( inj_on_c_c @ F @ A )
=> ( ( inj_on_c_a @ F2 @ ( image_c_c @ F @ A ) )
= ( inj_on_c_a @ ( comp_c_a_c @ F2 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_80_comp__inj__on__iff,axiom,
! [F: a > b,A: set_a,F2: b > c] :
( ( inj_on_a_b @ F @ A )
=> ( ( inj_on_b_c @ F2 @ ( image_a_b @ F @ A ) )
= ( inj_on_a_c @ ( comp_b_c_a @ F2 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_81_comp__inj__on__iff,axiom,
! [F: a > b,A: set_a,F2: b > a] :
( ( inj_on_a_b @ F @ A )
=> ( ( inj_on_b_a @ F2 @ ( image_a_b @ F @ A ) )
= ( inj_on_a_a @ ( comp_b_a_a @ F2 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_82_comp__inj__on__iff,axiom,
! [F: a > b,A: set_a,F2: b > b] :
( ( inj_on_a_b @ F @ A )
=> ( ( inj_on_b_b @ F2 @ ( image_a_b @ F @ A ) )
= ( inj_on_a_b @ ( comp_b_b_a @ F2 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_83_comp__inj__on__iff,axiom,
! [F: c > b,A: set_c,F2: b > a] :
( ( inj_on_c_b @ F @ A )
=> ( ( inj_on_b_a @ F2 @ ( image_c_b @ F @ A ) )
= ( inj_on_c_a @ ( comp_b_a_c @ F2 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_84_comp__inj__on__iff,axiom,
! [F: c > b,A: set_c,F2: b > b] :
( ( inj_on_c_b @ F @ A )
=> ( ( inj_on_b_b @ F2 @ ( image_c_b @ F @ A ) )
= ( inj_on_c_b @ ( comp_b_b_c @ F2 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_85_comp__inj__on__iff,axiom,
! [F: c > a,A: set_c,F2: a > b] :
( ( inj_on_c_a @ F @ A )
=> ( ( inj_on_a_b @ F2 @ ( image_c_a @ F @ A ) )
= ( inj_on_c_b @ ( comp_a_b_c @ F2 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_86_range__ex1__eq,axiom,
! [F: c > b,B2: b] :
( ( inj_on_c_b @ F @ top_top_set_c )
=> ( ( member_b @ B2 @ ( image_c_b @ F @ top_top_set_c ) )
= ( ? [X2: c] :
( ( B2
= ( F @ X2 ) )
& ! [Y: c] :
( ( B2
= ( F @ Y ) )
=> ( Y = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_87_range__ex1__eq,axiom,
! [F: c > a,B2: a] :
( ( inj_on_c_a @ F @ top_top_set_c )
=> ( ( member_a @ B2 @ ( image_c_a @ F @ top_top_set_c ) )
= ( ? [X2: c] :
( ( B2
= ( F @ X2 ) )
& ! [Y: c] :
( ( B2
= ( F @ Y ) )
=> ( Y = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_88_range__ex1__eq,axiom,
! [F: a > b,B2: b] :
( ( inj_on_a_b @ F @ top_top_set_a )
=> ( ( member_b @ B2 @ ( image_a_b @ F @ top_top_set_a ) )
= ( ? [X2: a] :
( ( B2
= ( F @ X2 ) )
& ! [Y: a] :
( ( B2
= ( F @ Y ) )
=> ( Y = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_89_range__ex1__eq,axiom,
! [F: a > c,B2: c] :
( ( inj_on_a_c @ F @ top_top_set_a )
=> ( ( member_c @ B2 @ ( image_a_c @ F @ top_top_set_a ) )
= ( ? [X2: a] :
( ( B2
= ( F @ X2 ) )
& ! [Y: a] :
( ( B2
= ( F @ Y ) )
=> ( Y = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_90_range__ex1__eq,axiom,
! [F: a > a,B2: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ B2 @ ( image_a_a @ F @ top_top_set_a ) )
= ( ? [X2: a] :
( ( B2
= ( F @ X2 ) )
& ! [Y: a] :
( ( B2
= ( F @ Y ) )
=> ( Y = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_91_range__ex1__eq,axiom,
! [F: b > b,B2: b] :
( ( inj_on_b_b @ F @ top_top_set_b )
=> ( ( member_b @ B2 @ ( image_b_b @ F @ top_top_set_b ) )
= ( ? [X2: b] :
( ( B2
= ( F @ X2 ) )
& ! [Y: b] :
( ( B2
= ( F @ Y ) )
=> ( Y = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_92_range__ex1__eq,axiom,
! [F: b > a,B2: a] :
( ( inj_on_b_a @ F @ top_top_set_b )
=> ( ( member_a @ B2 @ ( image_b_a @ F @ top_top_set_b ) )
= ( ? [X2: b] :
( ( B2
= ( F @ X2 ) )
& ! [Y: b] :
( ( B2
= ( F @ Y ) )
=> ( Y = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_93_range__ex1__eq,axiom,
! [F: ( c > a ) > a,B2: a] :
( ( inj_on_c_a_a @ F @ top_top_set_c_a )
=> ( ( member_a @ B2 @ ( image_c_a_a @ F @ top_top_set_c_a ) )
= ( ? [X2: c > a] :
( ( B2
= ( F @ X2 ) )
& ! [Y: c > a] :
( ( B2
= ( F @ Y ) )
=> ( Y = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_94_inj__image__eq__iff,axiom,
! [F: c > b,A: set_c,B: set_c] :
( ( inj_on_c_b @ F @ top_top_set_c )
=> ( ( ( image_c_b @ F @ A )
= ( image_c_b @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_95_inj__image__eq__iff,axiom,
! [F: c > a,A: set_c,B: set_c] :
( ( inj_on_c_a @ F @ top_top_set_c )
=> ( ( ( image_c_a @ F @ A )
= ( image_c_a @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_96_inj__image__eq__iff,axiom,
! [F: a > b,A: set_a,B: set_a] :
( ( inj_on_a_b @ F @ top_top_set_a )
=> ( ( ( image_a_b @ F @ A )
= ( image_a_b @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_97_inj__image__eq__iff,axiom,
! [F: a > c,A: set_a,B: set_a] :
( ( inj_on_a_c @ F @ top_top_set_a )
=> ( ( ( image_a_c @ F @ A )
= ( image_a_c @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_98_inj__image__eq__iff,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ( image_a_a @ F @ A )
= ( image_a_a @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_99_inj__image__eq__iff,axiom,
! [F: b > b,A: set_b,B: set_b] :
( ( inj_on_b_b @ F @ top_top_set_b )
=> ( ( ( image_b_b @ F @ A )
= ( image_b_b @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_100_inj__image__mem__iff,axiom,
! [F: c > b,A2: c,A: set_c] :
( ( inj_on_c_b @ F @ top_top_set_c )
=> ( ( member_b @ ( F @ A2 ) @ ( image_c_b @ F @ A ) )
= ( member_c @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_101_inj__image__mem__iff,axiom,
! [F: c > a,A2: c,A: set_c] :
( ( inj_on_c_a @ F @ top_top_set_c )
=> ( ( member_a @ ( F @ A2 ) @ ( image_c_a @ F @ A ) )
= ( member_c @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_102_inj__image__mem__iff,axiom,
! [F: a > b,A2: a,A: set_a] :
( ( inj_on_a_b @ F @ top_top_set_a )
=> ( ( member_b @ ( F @ A2 ) @ ( image_a_b @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_103_inj__image__mem__iff,axiom,
! [F: a > c,A2: a,A: set_a] :
( ( inj_on_a_c @ F @ top_top_set_a )
=> ( ( member_c @ ( F @ A2 ) @ ( image_a_c @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_104_inj__image__mem__iff,axiom,
! [F: a > a,A2: a,A: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ ( F @ A2 ) @ ( image_a_a @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_105_inj__image__mem__iff,axiom,
! [F: b > b,A2: b,A: set_b] :
( ( inj_on_b_b @ F @ top_top_set_b )
=> ( ( member_b @ ( F @ A2 ) @ ( image_b_b @ F @ A ) )
= ( member_b @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_106_inj__image__mem__iff,axiom,
! [F: b > a,A2: b,A: set_b] :
( ( inj_on_b_a @ F @ top_top_set_b )
=> ( ( member_a @ ( F @ A2 ) @ ( image_b_a @ F @ A ) )
= ( member_b @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_107_inj__image__mem__iff,axiom,
! [F: ( c > a ) > a,A2: c > a,A: set_c_a] :
( ( inj_on_c_a_a @ F @ top_top_set_c_a )
=> ( ( member_a @ ( F @ A2 ) @ ( image_c_a_a @ F @ A ) )
= ( member_c_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_108_comp__surj,axiom,
! [F: c > c,G: c > c] :
( ( ( image_c_c @ F @ top_top_set_c )
= top_top_set_c )
=> ( ( ( image_c_c @ G @ top_top_set_c )
= top_top_set_c )
=> ( ( image_c_c @ ( comp_c_c_c @ G @ F ) @ top_top_set_c )
= top_top_set_c ) ) ) ).
% comp_surj
thf(fact_109_comp__surj,axiom,
! [F: c > c,G: c > a] :
( ( ( image_c_c @ F @ top_top_set_c )
= top_top_set_c )
=> ( ( ( image_c_a @ G @ top_top_set_c )
= top_top_set_a )
=> ( ( image_c_a @ ( comp_c_a_c @ G @ F ) @ top_top_set_c )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_110_comp__surj,axiom,
! [F: c > c,G: c > b] :
( ( ( image_c_c @ F @ top_top_set_c )
= top_top_set_c )
=> ( ( ( image_c_b @ G @ top_top_set_c )
= top_top_set_b )
=> ( ( image_c_b @ ( comp_c_b_c @ G @ F ) @ top_top_set_c )
= top_top_set_b ) ) ) ).
% comp_surj
thf(fact_111_comp__surj,axiom,
! [F: c > a,G: a > c] :
( ( ( image_c_a @ F @ top_top_set_c )
= top_top_set_a )
=> ( ( ( image_a_c @ G @ top_top_set_a )
= top_top_set_c )
=> ( ( image_c_c @ ( comp_a_c_c @ G @ F ) @ top_top_set_c )
= top_top_set_c ) ) ) ).
% comp_surj
thf(fact_112_comp__surj,axiom,
! [F: c > a,G: a > a] :
( ( ( image_c_a @ F @ top_top_set_c )
= top_top_set_a )
=> ( ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a )
=> ( ( image_c_a @ ( comp_a_a_c @ G @ F ) @ top_top_set_c )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_113_comp__surj,axiom,
! [F: c > a,G: a > b] :
( ( ( image_c_a @ F @ top_top_set_c )
= top_top_set_a )
=> ( ( ( image_a_b @ G @ top_top_set_a )
= top_top_set_b )
=> ( ( image_c_b @ ( comp_a_b_c @ G @ F ) @ top_top_set_c )
= top_top_set_b ) ) ) ).
% comp_surj
thf(fact_114_comp__surj,axiom,
! [F: c > b,G: b > c] :
( ( ( image_c_b @ F @ top_top_set_c )
= top_top_set_b )
=> ( ( ( image_b_c @ G @ top_top_set_b )
= top_top_set_c )
=> ( ( image_c_c @ ( comp_b_c_c @ G @ F ) @ top_top_set_c )
= top_top_set_c ) ) ) ).
% comp_surj
thf(fact_115_comp__surj,axiom,
! [F: c > b,G: b > a] :
( ( ( image_c_b @ F @ top_top_set_c )
= top_top_set_b )
=> ( ( ( image_b_a @ G @ top_top_set_b )
= top_top_set_a )
=> ( ( image_c_a @ ( comp_b_a_c @ G @ F ) @ top_top_set_c )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_116_comp__surj,axiom,
! [F: c > b,G: b > b] :
( ( ( image_c_b @ F @ top_top_set_c )
= top_top_set_b )
=> ( ( ( image_b_b @ G @ top_top_set_b )
= top_top_set_b )
=> ( ( image_c_b @ ( comp_b_b_c @ G @ F ) @ top_top_set_c )
= top_top_set_b ) ) ) ).
% comp_surj
thf(fact_117_comp__surj,axiom,
! [F: a > c,G: c > c] :
( ( ( image_a_c @ F @ top_top_set_a )
= top_top_set_c )
=> ( ( ( image_c_c @ G @ top_top_set_c )
= top_top_set_c )
=> ( ( image_a_c @ ( comp_c_c_a @ G @ F ) @ top_top_set_a )
= top_top_set_c ) ) ) ).
% comp_surj
thf(fact_118_sup_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_119_sup_Oright__idem,axiom,
! [A2: set_c,B2: set_c] :
( ( sup_sup_set_c @ ( sup_sup_set_c @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_c @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_120_sup__left__idem,axiom,
! [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
= ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_121_sup__left__idem,axiom,
! [X3: set_c,Y2: set_c] :
( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ X3 @ Y2 ) )
= ( sup_sup_set_c @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_122_sup_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_123_sup_Oleft__idem,axiom,
! [A2: set_c,B2: set_c] :
( ( sup_sup_set_c @ A2 @ ( sup_sup_set_c @ A2 @ B2 ) )
= ( sup_sup_set_c @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_124_sup__idem,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_125_sup__idem,axiom,
! [X3: set_c] :
( ( sup_sup_set_c @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_126_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_127_sup_Oidem,axiom,
! [A2: set_c] :
( ( sup_sup_set_c @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_128_comp__apply,axiom,
( comp_a_b_c
= ( ^ [F3: a > b,G2: c > a,X2: c] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_129_comp__apply,axiom,
( comp_c_b_a
= ( ^ [F3: c > b,G2: a > c,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_130_comp__apply,axiom,
( comp_a_b_a
= ( ^ [F3: a > b,G2: a > a,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_131_comp__apply,axiom,
( comp_b_b_a
= ( ^ [F3: b > b,G2: a > b,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_132_sup__left__commute,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_133_sup__left__commute,axiom,
! [X3: set_c,Y2: set_c,Z: set_c] :
( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) )
= ( sup_sup_set_c @ Y2 @ ( sup_sup_set_c @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_134_sup_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_135_sup_Oleft__commute,axiom,
! [B2: set_c,A2: set_c,C: set_c] :
( ( sup_sup_set_c @ B2 @ ( sup_sup_set_c @ A2 @ C ) )
= ( sup_sup_set_c @ A2 @ ( sup_sup_set_c @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_136_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_137_sup__commute,axiom,
( sup_sup_set_c
= ( ^ [X2: set_c,Y: set_c] : ( sup_sup_set_c @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_138_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_139_sup_Ocommute,axiom,
( sup_sup_set_c
= ( ^ [A3: set_c,B3: set_c] : ( sup_sup_set_c @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_140_sup__assoc,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_141_sup__assoc,axiom,
! [X3: set_c,Y2: set_c,Z: set_c] :
( ( sup_sup_set_c @ ( sup_sup_set_c @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_142_sup_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_143_sup_Oassoc,axiom,
! [A2: set_c,B2: set_c,C: set_c] :
( ( sup_sup_set_c @ ( sup_sup_set_c @ A2 @ B2 ) @ C )
= ( sup_sup_set_c @ A2 @ ( sup_sup_set_c @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_144_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_145_boolean__algebra__cancel_Osup2,axiom,
! [B: set_c,K: set_c,B2: set_c,A2: set_c] :
( ( B
= ( sup_sup_set_c @ K @ B2 ) )
=> ( ( sup_sup_set_c @ A2 @ B )
= ( sup_sup_set_c @ K @ ( sup_sup_set_c @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_146_boolean__algebra__cancel_Osup1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_147_boolean__algebra__cancel_Osup1,axiom,
! [A: set_c,K: set_c,A2: set_c,B2: set_c] :
( ( A
= ( sup_sup_set_c @ K @ A2 ) )
=> ( ( sup_sup_set_c @ A @ B2 )
= ( sup_sup_set_c @ K @ ( sup_sup_set_c @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_148_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_149_inf__sup__aci_I5_J,axiom,
( sup_sup_set_c
= ( ^ [X2: set_c,Y: set_c] : ( sup_sup_set_c @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_150_inf__sup__aci_I6_J,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_151_inf__sup__aci_I6_J,axiom,
! [X3: set_c,Y2: set_c,Z: set_c] :
( ( sup_sup_set_c @ ( sup_sup_set_c @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_152_inf__sup__aci_I7_J,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_153_inf__sup__aci_I7_J,axiom,
! [X3: set_c,Y2: set_c,Z: set_c] :
( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) )
= ( sup_sup_set_c @ Y2 @ ( sup_sup_set_c @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_154_inf__sup__aci_I8_J,axiom,
! [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
= ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_155_inf__sup__aci_I8_J,axiom,
! [X3: set_c,Y2: set_c] :
( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ X3 @ Y2 ) )
= ( sup_sup_set_c @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_156_fun_Omap__comp,axiom,
! [G: c > b,F: a > c,V: c > a] :
( ( comp_c_b_c @ G @ ( comp_a_c_c @ F @ V ) )
= ( comp_a_b_c @ ( comp_c_b_a @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_157_fun_Omap__comp,axiom,
! [G: b > b,F: a > b,V: c > a] :
( ( comp_b_b_c @ G @ ( comp_a_b_c @ F @ V ) )
= ( comp_a_b_c @ ( comp_b_b_a @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_158_fun_Omap__comp,axiom,
! [G: a > b,F: c > a,V: c > c] :
( ( comp_a_b_c @ G @ ( comp_c_a_c @ F @ V ) )
= ( comp_c_b_c @ ( comp_a_b_c @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_159_fun_Omap__comp,axiom,
! [G: a > b,F: a > a,V: c > a] :
( ( comp_a_b_c @ G @ ( comp_a_a_c @ F @ V ) )
= ( comp_a_b_c @ ( comp_a_b_a @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_160_fun_Omap__comp,axiom,
! [G: c > b,F: c > c,V: a > c] :
( ( comp_c_b_a @ G @ ( comp_c_c_a @ F @ V ) )
= ( comp_c_b_a @ ( comp_c_b_c @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_161_fun_Omap__comp,axiom,
! [G: c > b,F: a > c,V: a > a] :
( ( comp_c_b_a @ G @ ( comp_a_c_a @ F @ V ) )
= ( comp_a_b_a @ ( comp_c_b_a @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_162_fun_Omap__comp,axiom,
! [G: c > b,F: b > c,V: a > b] :
( ( comp_c_b_a @ G @ ( comp_b_c_a @ F @ V ) )
= ( comp_b_b_a @ ( comp_c_b_b @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_163_fun_Omap__comp,axiom,
! [G: a > b,F: c > a,V: a > c] :
( ( comp_a_b_a @ G @ ( comp_c_a_a @ F @ V ) )
= ( comp_c_b_a @ ( comp_a_b_c @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_164_fun_Omap__comp,axiom,
! [G: a > b,F: a > a,V: a > a] :
( ( comp_a_b_a @ G @ ( comp_a_a_a @ F @ V ) )
= ( comp_a_b_a @ ( comp_a_b_a @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_165_fun_Omap__comp,axiom,
! [G: a > b,F: b > a,V: a > b] :
( ( comp_a_b_a @ G @ ( comp_b_a_a @ F @ V ) )
= ( comp_b_b_a @ ( comp_a_b_b @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_166_comp__eq__dest__lhs,axiom,
! [A2: a > b,B2: c > a,C: c > b,V: c] :
( ( ( comp_a_b_c @ A2 @ B2 )
= C )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_167_comp__eq__dest__lhs,axiom,
! [A2: c > b,B2: a > c,C: a > b,V: a] :
( ( ( comp_c_b_a @ A2 @ B2 )
= C )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_168_comp__eq__dest__lhs,axiom,
! [A2: a > b,B2: a > a,C: a > b,V: a] :
( ( ( comp_a_b_a @ A2 @ B2 )
= C )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_169_comp__eq__dest__lhs,axiom,
! [A2: b > b,B2: a > b,C: a > b,V: a] :
( ( ( comp_b_b_a @ A2 @ B2 )
= C )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_170_comp__eq__elim,axiom,
! [A2: a > b,B2: c > a,C: a > b,D: c > a] :
( ( ( comp_a_b_c @ A2 @ B2 )
= ( comp_a_b_c @ C @ D ) )
=> ! [V2: c] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_171_comp__eq__elim,axiom,
! [A2: c > b,B2: a > c,C: c > b,D: a > c] :
( ( ( comp_c_b_a @ A2 @ B2 )
= ( comp_c_b_a @ C @ D ) )
=> ! [V2: a] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_172_comp__eq__elim,axiom,
! [A2: c > b,B2: a > c,C: a > b,D: a > a] :
( ( ( comp_c_b_a @ A2 @ B2 )
= ( comp_a_b_a @ C @ D ) )
=> ! [V2: a] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_173_comp__eq__elim,axiom,
! [A2: c > b,B2: a > c,C: b > b,D: a > b] :
( ( ( comp_c_b_a @ A2 @ B2 )
= ( comp_b_b_a @ C @ D ) )
=> ! [V2: a] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_174_comp__eq__elim,axiom,
! [A2: a > b,B2: a > a,C: c > b,D: a > c] :
( ( ( comp_a_b_a @ A2 @ B2 )
= ( comp_c_b_a @ C @ D ) )
=> ! [V2: a] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_175_comp__eq__elim,axiom,
! [A2: a > b,B2: a > a,C: a > b,D: a > a] :
( ( ( comp_a_b_a @ A2 @ B2 )
= ( comp_a_b_a @ C @ D ) )
=> ! [V2: a] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_176_comp__eq__elim,axiom,
! [A2: a > b,B2: a > a,C: b > b,D: a > b] :
( ( ( comp_a_b_a @ A2 @ B2 )
= ( comp_b_b_a @ C @ D ) )
=> ! [V2: a] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_177_comp__eq__elim,axiom,
! [A2: b > b,B2: a > b,C: c > b,D: a > c] :
( ( ( comp_b_b_a @ A2 @ B2 )
= ( comp_c_b_a @ C @ D ) )
=> ! [V2: a] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_178_comp__eq__elim,axiom,
! [A2: b > b,B2: a > b,C: a > b,D: a > a] :
( ( ( comp_b_b_a @ A2 @ B2 )
= ( comp_a_b_a @ C @ D ) )
=> ! [V2: a] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_179_comp__eq__elim,axiom,
! [A2: b > b,B2: a > b,C: b > b,D: a > b] :
( ( ( comp_b_b_a @ A2 @ B2 )
= ( comp_b_b_a @ C @ D ) )
=> ! [V2: a] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_180_comp__eq__dest,axiom,
! [A2: a > b,B2: c > a,C: a > b,D: c > a,V: c] :
( ( ( comp_a_b_c @ A2 @ B2 )
= ( comp_a_b_c @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_181_comp__eq__dest,axiom,
! [A2: c > b,B2: a > c,C: c > b,D: a > c,V: a] :
( ( ( comp_c_b_a @ A2 @ B2 )
= ( comp_c_b_a @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_182_comp__eq__dest,axiom,
! [A2: c > b,B2: a > c,C: a > b,D: a > a,V: a] :
( ( ( comp_c_b_a @ A2 @ B2 )
= ( comp_a_b_a @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_183_comp__eq__dest,axiom,
! [A2: c > b,B2: a > c,C: b > b,D: a > b,V: a] :
( ( ( comp_c_b_a @ A2 @ B2 )
= ( comp_b_b_a @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_184_comp__eq__dest,axiom,
! [A2: a > b,B2: a > a,C: c > b,D: a > c,V: a] :
( ( ( comp_a_b_a @ A2 @ B2 )
= ( comp_c_b_a @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_185_comp__eq__dest,axiom,
! [A2: a > b,B2: a > a,C: a > b,D: a > a,V: a] :
( ( ( comp_a_b_a @ A2 @ B2 )
= ( comp_a_b_a @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_186_comp__eq__dest,axiom,
! [A2: a > b,B2: a > a,C: b > b,D: a > b,V: a] :
( ( ( comp_a_b_a @ A2 @ B2 )
= ( comp_b_b_a @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_187_comp__eq__dest,axiom,
! [A2: b > b,B2: a > b,C: c > b,D: a > c,V: a] :
( ( ( comp_b_b_a @ A2 @ B2 )
= ( comp_c_b_a @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_188_comp__eq__dest,axiom,
! [A2: b > b,B2: a > b,C: a > b,D: a > a,V: a] :
( ( ( comp_b_b_a @ A2 @ B2 )
= ( comp_a_b_a @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_189_comp__eq__dest,axiom,
! [A2: b > b,B2: a > b,C: b > b,D: a > b,V: a] :
( ( ( comp_b_b_a @ A2 @ B2 )
= ( comp_b_b_a @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_190_comp__assoc,axiom,
! [F: a > b,G: c > a,H: c > c] :
( ( comp_c_b_c @ ( comp_a_b_c @ F @ G ) @ H )
= ( comp_a_b_c @ F @ ( comp_c_a_c @ G @ H ) ) ) ).
% comp_assoc
thf(fact_191_comp__assoc,axiom,
! [F: c > b,G: a > c,H: c > a] :
( ( comp_a_b_c @ ( comp_c_b_a @ F @ G ) @ H )
= ( comp_c_b_c @ F @ ( comp_a_c_c @ G @ H ) ) ) ).
% comp_assoc
thf(fact_192_comp__assoc,axiom,
! [F: a > b,G: a > a,H: c > a] :
( ( comp_a_b_c @ ( comp_a_b_a @ F @ G ) @ H )
= ( comp_a_b_c @ F @ ( comp_a_a_c @ G @ H ) ) ) ).
% comp_assoc
thf(fact_193_comp__assoc,axiom,
! [F: b > b,G: a > b,H: c > a] :
( ( comp_a_b_c @ ( comp_b_b_a @ F @ G ) @ H )
= ( comp_b_b_c @ F @ ( comp_a_b_c @ G @ H ) ) ) ).
% comp_assoc
thf(fact_194_comp__assoc,axiom,
! [F: c > b,G: c > c,H: a > c] :
( ( comp_c_b_a @ ( comp_c_b_c @ F @ G ) @ H )
= ( comp_c_b_a @ F @ ( comp_c_c_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_195_comp__assoc,axiom,
! [F: b > b,G: c > b,H: a > c] :
( ( comp_c_b_a @ ( comp_b_b_c @ F @ G ) @ H )
= ( comp_b_b_a @ F @ ( comp_c_b_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_196_comp__assoc,axiom,
! [F: a > b,G: c > a,H: a > c] :
( ( comp_c_b_a @ ( comp_a_b_c @ F @ G ) @ H )
= ( comp_a_b_a @ F @ ( comp_c_a_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_197_comp__assoc,axiom,
! [F: c > b,G: a > c,H: a > a] :
( ( comp_a_b_a @ ( comp_c_b_a @ F @ G ) @ H )
= ( comp_c_b_a @ F @ ( comp_a_c_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_198_comp__assoc,axiom,
! [F: a > b,G: a > a,H: a > a] :
( ( comp_a_b_a @ ( comp_a_b_a @ F @ G ) @ H )
= ( comp_a_b_a @ F @ ( comp_a_a_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_199_comp__assoc,axiom,
! [F: b > b,G: a > b,H: a > a] :
( ( comp_a_b_a @ ( comp_b_b_a @ F @ G ) @ H )
= ( comp_b_b_a @ F @ ( comp_a_b_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_200_comp__def,axiom,
( comp_a_b_c
= ( ^ [F3: a > b,G2: c > a,X2: c] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_201_comp__def,axiom,
( comp_c_b_a
= ( ^ [F3: c > b,G2: a > c,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_202_comp__def,axiom,
( comp_a_b_a
= ( ^ [F3: a > b,G2: a > a,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_203_comp__def,axiom,
( comp_b_b_a
= ( ^ [F3: b > b,G2: a > b,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_204_inj__on__inverseI,axiom,
! [A: set_c,G: b > c,F: c > b] :
( ! [X: c] :
( ( member_c @ X @ A )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on_c_b @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_205_inj__on__inverseI,axiom,
! [A: set_c,G: a > c,F: c > a] :
( ! [X: c] :
( ( member_c @ X @ A )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on_c_a @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_206_inj__on__inverseI,axiom,
! [A: set_b,G: b > b,F: b > b] :
( ! [X: b] :
( ( member_b @ X @ A )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on_b_b @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_207_inj__on__inverseI,axiom,
! [A: set_a,G: b > a,F: a > b] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on_a_b @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_208_inj__on__inverseI,axiom,
! [A: set_a,G: c > a,F: a > c] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on_a_c @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_209_inj__on__inverseI,axiom,
! [A: set_a,G: a > a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on_a_a @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_210_inj__on__contraD,axiom,
! [F: c > b,A: set_c,X3: c,Y2: c] :
( ( inj_on_c_b @ F @ A )
=> ( ( X3 != Y2 )
=> ( ( member_c @ X3 @ A )
=> ( ( member_c @ Y2 @ A )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_211_inj__on__contraD,axiom,
! [F: c > a,A: set_c,X3: c,Y2: c] :
( ( inj_on_c_a @ F @ A )
=> ( ( X3 != Y2 )
=> ( ( member_c @ X3 @ A )
=> ( ( member_c @ Y2 @ A )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_212_inj__on__contraD,axiom,
! [F: b > b,A: set_b,X3: b,Y2: b] :
( ( inj_on_b_b @ F @ A )
=> ( ( X3 != Y2 )
=> ( ( member_b @ X3 @ A )
=> ( ( member_b @ Y2 @ A )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_213_inj__on__contraD,axiom,
! [F: a > b,A: set_a,X3: a,Y2: a] :
( ( inj_on_a_b @ F @ A )
=> ( ( X3 != Y2 )
=> ( ( member_a @ X3 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_214_inj__on__contraD,axiom,
! [F: a > c,A: set_a,X3: a,Y2: a] :
( ( inj_on_a_c @ F @ A )
=> ( ( X3 != Y2 )
=> ( ( member_a @ X3 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_215_inj__on__contraD,axiom,
! [F: a > a,A: set_a,X3: a,Y2: a] :
( ( inj_on_a_a @ F @ A )
=> ( ( X3 != Y2 )
=> ( ( member_a @ X3 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_216_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_217_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_218_inj__on__eq__iff,axiom,
! [F: c > b,A: set_c,X3: c,Y2: c] :
( ( inj_on_c_b @ F @ A )
=> ( ( member_c @ X3 @ A )
=> ( ( member_c @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_219_inj__on__eq__iff,axiom,
! [F: c > a,A: set_c,X3: c,Y2: c] :
( ( inj_on_c_a @ F @ A )
=> ( ( member_c @ X3 @ A )
=> ( ( member_c @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_220_inj__on__eq__iff,axiom,
! [F: b > b,A: set_b,X3: b,Y2: b] :
( ( inj_on_b_b @ F @ A )
=> ( ( member_b @ X3 @ A )
=> ( ( member_b @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_221_inj__on__eq__iff,axiom,
! [F: a > b,A: set_a,X3: a,Y2: a] :
( ( inj_on_a_b @ F @ A )
=> ( ( member_a @ X3 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_222_inj__on__eq__iff,axiom,
! [F: a > c,A: set_a,X3: a,Y2: a] :
( ( inj_on_a_c @ F @ A )
=> ( ( member_a @ X3 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_223_inj__on__eq__iff,axiom,
! [F: a > a,A: set_a,X3: a,Y2: a] :
( ( inj_on_a_a @ F @ A )
=> ( ( member_a @ X3 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_224_inj__on__cong,axiom,
! [A: set_c,F: c > b,G: c > b] :
( ! [A4: c] :
( ( member_c @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_c_b @ F @ A )
= ( inj_on_c_b @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_225_inj__on__cong,axiom,
! [A: set_c,F: c > a,G: c > a] :
( ! [A4: c] :
( ( member_c @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_c_a @ F @ A )
= ( inj_on_c_a @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_226_inj__on__cong,axiom,
! [A: set_b,F: b > b,G: b > b] :
( ! [A4: b] :
( ( member_b @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_b_b @ F @ A )
= ( inj_on_b_b @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_227_inj__on__cong,axiom,
! [A: set_a,F: a > b,G: a > b] :
( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_a_b @ F @ A )
= ( inj_on_a_b @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_228_inj__on__cong,axiom,
! [A: set_a,F: a > c,G: a > c] :
( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_a_c @ F @ A )
= ( inj_on_a_c @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_229_inj__on__cong,axiom,
! [A: set_a,F: a > a,G: a > a] :
( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_a_a @ F @ A )
= ( inj_on_a_a @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_230_inj__on__def,axiom,
( inj_on_a_b
= ( ^ [F3: a > b,A5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ! [Y: a] :
( ( member_a @ Y @ A5 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y ) )
=> ( X2 = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_231_inj__on__def,axiom,
( inj_on_c_b
= ( ^ [F3: c > b,A5: set_c] :
! [X2: c] :
( ( member_c @ X2 @ A5 )
=> ! [Y: c] :
( ( member_c @ Y @ A5 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y ) )
=> ( X2 = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_232_inj__on__def,axiom,
( inj_on_c_a
= ( ^ [F3: c > a,A5: set_c] :
! [X2: c] :
( ( member_c @ X2 @ A5 )
=> ! [Y: c] :
( ( member_c @ Y @ A5 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y ) )
=> ( X2 = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_233_inj__on__def,axiom,
( inj_on_b_b
= ( ^ [F3: b > b,A5: set_b] :
! [X2: b] :
( ( member_b @ X2 @ A5 )
=> ! [Y: b] :
( ( member_b @ Y @ A5 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y ) )
=> ( X2 = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_234_inj__on__def,axiom,
( inj_on_a_c
= ( ^ [F3: a > c,A5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ! [Y: a] :
( ( member_a @ Y @ A5 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y ) )
=> ( X2 = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_235_inj__on__def,axiom,
( inj_on_a_a
= ( ^ [F3: a > a,A5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ! [Y: a] :
( ( member_a @ Y @ A5 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y ) )
=> ( X2 = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_236_inj__onI,axiom,
! [A: set_c,F: c > b] :
( ! [X: c,Y3: c] :
( ( member_c @ X @ A )
=> ( ( member_c @ Y3 @ A )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) )
=> ( inj_on_c_b @ F @ A ) ) ).
% inj_onI
thf(fact_237_inj__onI,axiom,
! [A: set_c,F: c > a] :
( ! [X: c,Y3: c] :
( ( member_c @ X @ A )
=> ( ( member_c @ Y3 @ A )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) )
=> ( inj_on_c_a @ F @ A ) ) ).
% inj_onI
thf(fact_238_inj__onI,axiom,
! [A: set_b,F: b > b] :
( ! [X: b,Y3: b] :
( ( member_b @ X @ A )
=> ( ( member_b @ Y3 @ A )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) )
=> ( inj_on_b_b @ F @ A ) ) ).
% inj_onI
thf(fact_239_inj__onI,axiom,
! [A: set_a,F: a > b] :
( ! [X: a,Y3: a] :
( ( member_a @ X @ A )
=> ( ( member_a @ Y3 @ A )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) )
=> ( inj_on_a_b @ F @ A ) ) ).
% inj_onI
thf(fact_240_inj__onI,axiom,
! [A: set_a,F: a > c] :
( ! [X: a,Y3: a] :
( ( member_a @ X @ A )
=> ( ( member_a @ Y3 @ A )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) )
=> ( inj_on_a_c @ F @ A ) ) ).
% inj_onI
thf(fact_241_inj__onI,axiom,
! [A: set_a,F: a > a] :
( ! [X: a,Y3: a] :
( ( member_a @ X @ A )
=> ( ( member_a @ Y3 @ A )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) )
=> ( inj_on_a_a @ F @ A ) ) ).
% inj_onI
thf(fact_242_inj__onD,axiom,
! [F: c > b,A: set_c,X3: c,Y2: c] :
( ( inj_on_c_b @ F @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( ( member_c @ X3 @ A )
=> ( ( member_c @ Y2 @ A )
=> ( X3 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_243_inj__onD,axiom,
! [F: c > a,A: set_c,X3: c,Y2: c] :
( ( inj_on_c_a @ F @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( ( member_c @ X3 @ A )
=> ( ( member_c @ Y2 @ A )
=> ( X3 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_244_inj__onD,axiom,
! [F: b > b,A: set_b,X3: b,Y2: b] :
( ( inj_on_b_b @ F @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( ( member_b @ X3 @ A )
=> ( ( member_b @ Y2 @ A )
=> ( X3 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_245_inj__onD,axiom,
! [F: a > b,A: set_a,X3: a,Y2: a] :
( ( inj_on_a_b @ F @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( ( member_a @ X3 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( X3 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_246_inj__onD,axiom,
! [F: a > c,A: set_a,X3: a,Y2: a] :
( ( inj_on_a_c @ F @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( ( member_a @ X3 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( X3 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_247_inj__onD,axiom,
! [F: a > a,A: set_a,X3: a,Y2: a] :
( ( inj_on_a_a @ F @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( ( member_a @ X3 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( X3 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_248_surj__def,axiom,
! [F: c > c] :
( ( ( image_c_c @ F @ top_top_set_c )
= top_top_set_c )
= ( ! [Y: c] :
? [X2: c] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_249_surj__def,axiom,
! [F: c > a] :
( ( ( image_c_a @ F @ top_top_set_c )
= top_top_set_a )
= ( ! [Y: a] :
? [X2: c] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_250_surj__def,axiom,
! [F: c > b] :
( ( ( image_c_b @ F @ top_top_set_c )
= top_top_set_b )
= ( ! [Y: b] :
? [X2: c] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_251_surj__def,axiom,
! [F: a > c] :
( ( ( image_a_c @ F @ top_top_set_a )
= top_top_set_c )
= ( ! [Y: c] :
? [X2: a] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_252_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y: a] :
? [X2: a] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_253_surj__def,axiom,
! [F: a > b] :
( ( ( image_a_b @ F @ top_top_set_a )
= top_top_set_b )
= ( ! [Y: b] :
? [X2: a] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_254_surj__def,axiom,
! [F: b > c] :
( ( ( image_b_c @ F @ top_top_set_b )
= top_top_set_c )
= ( ! [Y: c] :
? [X2: b] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_255_surj__def,axiom,
! [F: b > a] :
( ( ( image_b_a @ F @ top_top_set_b )
= top_top_set_a )
= ( ! [Y: a] :
? [X2: b] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_256_surj__def,axiom,
! [F: b > b] :
( ( ( image_b_b @ F @ top_top_set_b )
= top_top_set_b )
= ( ! [Y: b] :
? [X2: b] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_257_surj__def,axiom,
! [F: c > c > a] :
( ( ( image_c_c_a @ F @ top_top_set_c )
= top_top_set_c_a )
= ( ! [Y: c > a] :
? [X2: c] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_258_surjI,axiom,
! [G: c > c,F: c > c] :
( ! [X: c] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_c_c @ G @ top_top_set_c )
= top_top_set_c ) ) ).
% surjI
thf(fact_259_surjI,axiom,
! [G: c > a,F: a > c] :
( ! [X: a] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_c_a @ G @ top_top_set_c )
= top_top_set_a ) ) ).
% surjI
thf(fact_260_surjI,axiom,
! [G: c > b,F: b > c] :
( ! [X: b] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_c_b @ G @ top_top_set_c )
= top_top_set_b ) ) ).
% surjI
thf(fact_261_surjI,axiom,
! [G: a > c,F: c > a] :
( ! [X: c] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_a_c @ G @ top_top_set_a )
= top_top_set_c ) ) ).
% surjI
thf(fact_262_surjI,axiom,
! [G: a > a,F: a > a] :
( ! [X: a] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_263_surjI,axiom,
! [G: a > b,F: b > a] :
( ! [X: b] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_a_b @ G @ top_top_set_a )
= top_top_set_b ) ) ).
% surjI
thf(fact_264_surjI,axiom,
! [G: b > c,F: c > b] :
( ! [X: c] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_b_c @ G @ top_top_set_b )
= top_top_set_c ) ) ).
% surjI
thf(fact_265_surjI,axiom,
! [G: b > a,F: a > b] :
( ! [X: a] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_b_a @ G @ top_top_set_b )
= top_top_set_a ) ) ).
% surjI
thf(fact_266_surjI,axiom,
! [G: b > b,F: b > b] :
( ! [X: b] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_b_b @ G @ top_top_set_b )
= top_top_set_b ) ) ).
% surjI
thf(fact_267_surjI,axiom,
! [G: c > c > a,F: ( c > a ) > c] :
( ! [X: c > a] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_c_c_a @ G @ top_top_set_c )
= top_top_set_c_a ) ) ).
% surjI
thf(fact_268_surjE,axiom,
! [F: c > c,Y2: c] :
( ( ( image_c_c @ F @ top_top_set_c )
= top_top_set_c )
=> ~ ! [X: c] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_269_surjE,axiom,
! [F: c > a,Y2: a] :
( ( ( image_c_a @ F @ top_top_set_c )
= top_top_set_a )
=> ~ ! [X: c] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_270_surjE,axiom,
! [F: c > b,Y2: b] :
( ( ( image_c_b @ F @ top_top_set_c )
= top_top_set_b )
=> ~ ! [X: c] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_271_surjE,axiom,
! [F: a > c,Y2: c] :
( ( ( image_a_c @ F @ top_top_set_a )
= top_top_set_c )
=> ~ ! [X: a] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_272_surjE,axiom,
! [F: a > a,Y2: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X: a] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_273_surjE,axiom,
! [F: a > b,Y2: b] :
( ( ( image_a_b @ F @ top_top_set_a )
= top_top_set_b )
=> ~ ! [X: a] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_274_surjE,axiom,
! [F: b > c,Y2: c] :
( ( ( image_b_c @ F @ top_top_set_b )
= top_top_set_c )
=> ~ ! [X: b] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_275_surjE,axiom,
! [F: b > a,Y2: a] :
( ( ( image_b_a @ F @ top_top_set_b )
= top_top_set_a )
=> ~ ! [X: b] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_276_surjE,axiom,
! [F: b > b,Y2: b] :
( ( ( image_b_b @ F @ top_top_set_b )
= top_top_set_b )
=> ~ ! [X: b] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_277_surjE,axiom,
! [F: c > c > a,Y2: c > a] :
( ( ( image_c_c_a @ F @ top_top_set_c )
= top_top_set_c_a )
=> ~ ! [X: c] :
( Y2
!= ( F @ X ) ) ) ).
% surjE
thf(fact_278_surjD,axiom,
! [F: c > c,Y2: c] :
( ( ( image_c_c @ F @ top_top_set_c )
= top_top_set_c )
=> ? [X: c] :
( Y2
= ( F @ X ) ) ) ).
% surjD
thf(fact_279_surjD,axiom,
! [F: c > a,Y2: a] :
( ( ( image_c_a @ F @ top_top_set_c )
= top_top_set_a )
=> ? [X: c] :
( Y2
= ( F @ X ) ) ) ).
% surjD
thf(fact_280_surjD,axiom,
! [F: c > b,Y2: b] :
( ( ( image_c_b @ F @ top_top_set_c )
= top_top_set_b )
=> ? [X: c] :
( Y2
= ( F @ X ) ) ) ).
% surjD
thf(fact_281_surjD,axiom,
! [F: a > c,Y2: c] :
( ( ( image_a_c @ F @ top_top_set_a )
= top_top_set_c )
=> ? [X: a] :
( Y2
= ( F @ X ) ) ) ).
% surjD
thf(fact_282_surjD,axiom,
! [F: a > a,Y2: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ? [X: a] :
( Y2
= ( F @ X ) ) ) ).
% surjD
thf(fact_283_surjD,axiom,
! [F: a > b,Y2: b] :
( ( ( image_a_b @ F @ top_top_set_a )
= top_top_set_b )
=> ? [X: a] :
( Y2
= ( F @ X ) ) ) ).
% surjD
thf(fact_284_surjD,axiom,
! [F: b > c,Y2: c] :
( ( ( image_b_c @ F @ top_top_set_b )
= top_top_set_c )
=> ? [X: b] :
( Y2
= ( F @ X ) ) ) ).
% surjD
thf(fact_285_surjD,axiom,
! [F: b > a,Y2: a] :
( ( ( image_b_a @ F @ top_top_set_b )
= top_top_set_a )
=> ? [X: b] :
( Y2
= ( F @ X ) ) ) ).
% surjD
thf(fact_286_surjD,axiom,
! [F: b > b,Y2: b] :
( ( ( image_b_b @ F @ top_top_set_b )
= top_top_set_b )
=> ? [X: b] :
( Y2
= ( F @ X ) ) ) ).
% surjD
thf(fact_287_surjD,axiom,
! [F: c > c > a,Y2: c > a] :
( ( ( image_c_c_a @ F @ top_top_set_c )
= top_top_set_c_a )
=> ? [X: c] :
( Y2
= ( F @ X ) ) ) ).
% surjD
thf(fact_288_image__eq__imp__comp,axiom,
! [F: c > c,A: set_c,G: a > c,B: set_a,H: c > b] :
( ( ( image_c_c @ F @ A )
= ( image_a_c @ G @ B ) )
=> ( ( image_c_b @ ( comp_c_b_c @ H @ F ) @ A )
= ( image_a_b @ ( comp_c_b_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_289_image__eq__imp__comp,axiom,
! [F: a > c,A: set_a,G: c > c,B: set_c,H: c > b] :
( ( ( image_a_c @ F @ A )
= ( image_c_c @ G @ B ) )
=> ( ( image_a_b @ ( comp_c_b_a @ H @ F ) @ A )
= ( image_c_b @ ( comp_c_b_c @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_290_image__eq__imp__comp,axiom,
! [F: a > c,A: set_a,G: a > c,B: set_a,H: c > b] :
( ( ( image_a_c @ F @ A )
= ( image_a_c @ G @ B ) )
=> ( ( image_a_b @ ( comp_c_b_a @ H @ F ) @ A )
= ( image_a_b @ ( comp_c_b_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_291_image__eq__imp__comp,axiom,
! [F: a > a,A: set_a,G: a > a,B: set_a,H: a > b] :
( ( ( image_a_a @ F @ A )
= ( image_a_a @ G @ B ) )
=> ( ( image_a_b @ ( comp_a_b_a @ H @ F ) @ A )
= ( image_a_b @ ( comp_a_b_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_292_image__eq__imp__comp,axiom,
! [F: a > a,A: set_a,G: c > a,B: set_c,H: a > b] :
( ( ( image_a_a @ F @ A )
= ( image_c_a @ G @ B ) )
=> ( ( image_a_b @ ( comp_a_b_a @ H @ F ) @ A )
= ( image_c_b @ ( comp_a_b_c @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_293_image__eq__imp__comp,axiom,
! [F: c > a,A: set_c,G: a > a,B: set_a,H: a > b] :
( ( ( image_c_a @ F @ A )
= ( image_a_a @ G @ B ) )
=> ( ( image_c_b @ ( comp_a_b_c @ H @ F ) @ A )
= ( image_a_b @ ( comp_a_b_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_294_image__eq__imp__comp,axiom,
! [F: c > a,A: set_c,G: c > a,B: set_c,H: a > a] :
( ( ( image_c_a @ F @ A )
= ( image_c_a @ G @ B ) )
=> ( ( image_c_a @ ( comp_a_a_c @ H @ F ) @ A )
= ( image_c_a @ ( comp_a_a_c @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_295_image__eq__imp__comp,axiom,
! [F: c > a,A: set_c,G: c > a,B: set_c,H: a > b] :
( ( ( image_c_a @ F @ A )
= ( image_c_a @ G @ B ) )
=> ( ( image_c_b @ ( comp_a_b_c @ H @ F ) @ A )
= ( image_c_b @ ( comp_a_b_c @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_296_image__eq__imp__comp,axiom,
! [F: c > b,A: set_c,G: c > b,B: set_c,H: b > a] :
( ( ( image_c_b @ F @ A )
= ( image_c_b @ G @ B ) )
=> ( ( image_c_a @ ( comp_b_a_c @ H @ F ) @ A )
= ( image_c_a @ ( comp_b_a_c @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_297_image__eq__imp__comp,axiom,
! [F: c > b,A: set_c,G: c > b,B: set_c,H: b > b] :
( ( ( image_c_b @ F @ A )
= ( image_c_b @ G @ B ) )
=> ( ( image_c_b @ ( comp_b_b_c @ H @ F ) @ A )
= ( image_c_b @ ( comp_b_b_c @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_298_image__comp,axiom,
! [F: a > a,G: c > a,R: set_c] :
( ( image_a_a @ F @ ( image_c_a @ G @ R ) )
= ( image_c_a @ ( comp_a_a_c @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_299_image__comp,axiom,
! [F: b > a,G: c > b,R: set_c] :
( ( image_b_a @ F @ ( image_c_b @ G @ R ) )
= ( image_c_a @ ( comp_b_a_c @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_300_image__comp,axiom,
! [F: b > b,G: c > b,R: set_c] :
( ( image_b_b @ F @ ( image_c_b @ G @ R ) )
= ( image_c_b @ ( comp_b_b_c @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_301_image__comp,axiom,
! [F: b > b,G: a > b,R: set_a] :
( ( image_b_b @ F @ ( image_a_b @ G @ R ) )
= ( image_a_b @ ( comp_b_b_a @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_302_image__comp,axiom,
! [F: c > a,G: c > c,R: set_c] :
( ( image_c_a @ F @ ( image_c_c @ G @ R ) )
= ( image_c_a @ ( comp_c_a_c @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_303_image__comp,axiom,
! [F: c > b,G: c > c,R: set_c] :
( ( image_c_b @ F @ ( image_c_c @ G @ R ) )
= ( image_c_b @ ( comp_c_b_c @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_304_image__comp,axiom,
! [F: c > b,G: a > c,R: set_a] :
( ( image_c_b @ F @ ( image_a_c @ G @ R ) )
= ( image_a_b @ ( comp_c_b_a @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_305_image__comp,axiom,
! [F: a > b,G: a > a,R: set_a] :
( ( image_a_b @ F @ ( image_a_a @ G @ R ) )
= ( image_a_b @ ( comp_a_b_a @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_306_image__comp,axiom,
! [F: a > b,G: c > a,R: set_c] :
( ( image_a_b @ F @ ( image_c_a @ G @ R ) )
= ( image_c_b @ ( comp_a_b_c @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_307_inj__on__image__iff,axiom,
! [A: set_c,G: c > b,F: c > c] :
( ! [X: c] :
( ( member_c @ X @ A )
=> ! [Xa: c] :
( ( member_c @ Xa @ A )
=> ( ( ( G @ ( F @ X ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_c_c @ F @ A )
=> ( ( inj_on_c_b @ G @ ( image_c_c @ F @ A ) )
= ( inj_on_c_b @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_308_inj__on__image__iff,axiom,
! [A: set_c,G: c > a,F: c > c] :
( ! [X: c] :
( ( member_c @ X @ A )
=> ! [Xa: c] :
( ( member_c @ Xa @ A )
=> ( ( ( G @ ( F @ X ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_c_c @ F @ A )
=> ( ( inj_on_c_a @ G @ ( image_c_c @ F @ A ) )
= ( inj_on_c_a @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_309_inj__on__image__iff,axiom,
! [A: set_b,G: b > b,F: b > b] :
( ! [X: b] :
( ( member_b @ X @ A )
=> ! [Xa: b] :
( ( member_b @ Xa @ A )
=> ( ( ( G @ ( F @ X ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_b_b @ F @ A )
=> ( ( inj_on_b_b @ G @ ( image_b_b @ F @ A ) )
= ( inj_on_b_b @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_310_inj__on__image__iff,axiom,
! [A: set_a,G: a > b,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ( G @ ( F @ X ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A )
=> ( ( inj_on_a_b @ G @ ( image_a_a @ F @ A ) )
= ( inj_on_a_b @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_311_inj__on__image__iff,axiom,
! [A: set_a,G: a > c,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ( G @ ( F @ X ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A )
=> ( ( inj_on_a_c @ G @ ( image_a_a @ F @ A ) )
= ( inj_on_a_c @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_312_inj__on__image__iff,axiom,
! [A: set_a,G: a > a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ( G @ ( F @ X ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A )
=> ( ( inj_on_a_a @ G @ ( image_a_a @ F @ A ) )
= ( inj_on_a_a @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_313_inj__def,axiom,
! [F: c > b] :
( ( inj_on_c_b @ F @ top_top_set_c )
= ( ! [X2: c,Y: c] :
( ( ( F @ X2 )
= ( F @ Y ) )
=> ( X2 = Y ) ) ) ) ).
% inj_def
thf(fact_314_inj__def,axiom,
! [F: c > a] :
( ( inj_on_c_a @ F @ top_top_set_c )
= ( ! [X2: c,Y: c] :
( ( ( F @ X2 )
= ( F @ Y ) )
=> ( X2 = Y ) ) ) ) ).
% inj_def
thf(fact_315_inj__def,axiom,
! [F: a > b] :
( ( inj_on_a_b @ F @ top_top_set_a )
= ( ! [X2: a,Y: a] :
( ( ( F @ X2 )
= ( F @ Y ) )
=> ( X2 = Y ) ) ) ) ).
% inj_def
thf(fact_316_inj__def,axiom,
! [F: a > c] :
( ( inj_on_a_c @ F @ top_top_set_a )
= ( ! [X2: a,Y: a] :
( ( ( F @ X2 )
= ( F @ Y ) )
=> ( X2 = Y ) ) ) ) ).
% inj_def
thf(fact_317_inj__def,axiom,
! [F: a > a] :
( ( inj_on_a_a @ F @ top_top_set_a )
= ( ! [X2: a,Y: a] :
( ( ( F @ X2 )
= ( F @ Y ) )
=> ( X2 = Y ) ) ) ) ).
% inj_def
thf(fact_318_inj__def,axiom,
! [F: b > b] :
( ( inj_on_b_b @ F @ top_top_set_b )
= ( ! [X2: b,Y: b] :
( ( ( F @ X2 )
= ( F @ Y ) )
=> ( X2 = Y ) ) ) ) ).
% inj_def
thf(fact_319_inj__eq,axiom,
! [F: c > b,X3: c,Y2: c] :
( ( inj_on_c_b @ F @ top_top_set_c )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% inj_eq
thf(fact_320_inj__eq,axiom,
! [F: c > a,X3: c,Y2: c] :
( ( inj_on_c_a @ F @ top_top_set_c )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% inj_eq
thf(fact_321_inj__eq,axiom,
! [F: a > b,X3: a,Y2: a] :
( ( inj_on_a_b @ F @ top_top_set_a )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% inj_eq
thf(fact_322_inj__eq,axiom,
! [F: a > c,X3: a,Y2: a] :
( ( inj_on_a_c @ F @ top_top_set_a )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% inj_eq
thf(fact_323_inj__eq,axiom,
! [F: a > a,X3: a,Y2: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% inj_eq
thf(fact_324_inj__eq,axiom,
! [F: b > b,X3: b,Y2: b] :
( ( inj_on_b_b @ F @ top_top_set_b )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% inj_eq
thf(fact_325_injI,axiom,
! [F: c > b] :
( ! [X: c,Y3: c] :
( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) )
=> ( inj_on_c_b @ F @ top_top_set_c ) ) ).
% injI
thf(fact_326_injI,axiom,
! [F: c > a] :
( ! [X: c,Y3: c] :
( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) )
=> ( inj_on_c_a @ F @ top_top_set_c ) ) ).
% injI
thf(fact_327_injI,axiom,
! [F: a > b] :
( ! [X: a,Y3: a] :
( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) )
=> ( inj_on_a_b @ F @ top_top_set_a ) ) ).
% injI
thf(fact_328_injI,axiom,
! [F: a > c] :
( ! [X: a,Y3: a] :
( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) )
=> ( inj_on_a_c @ F @ top_top_set_a ) ) ).
% injI
thf(fact_329_injI,axiom,
! [F: a > a] :
( ! [X: a,Y3: a] :
( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) )
=> ( inj_on_a_a @ F @ top_top_set_a ) ) ).
% injI
thf(fact_330_injI,axiom,
! [F: b > b] :
( ! [X: b,Y3: b] :
( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) )
=> ( inj_on_b_b @ F @ top_top_set_b ) ) ).
% injI
thf(fact_331_injD,axiom,
! [F: c > b,X3: c,Y2: c] :
( ( inj_on_c_b @ F @ top_top_set_c )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) ).
% injD
thf(fact_332_injD,axiom,
! [F: c > a,X3: c,Y2: c] :
( ( inj_on_c_a @ F @ top_top_set_c )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) ).
% injD
thf(fact_333_injD,axiom,
! [F: a > b,X3: a,Y2: a] :
( ( inj_on_a_b @ F @ top_top_set_a )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) ).
% injD
thf(fact_334_injD,axiom,
! [F: a > c,X3: a,Y2: a] :
( ( inj_on_a_c @ F @ top_top_set_a )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) ).
% injD
thf(fact_335_injD,axiom,
! [F: a > a,X3: a,Y2: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) ).
% injD
thf(fact_336_injD,axiom,
! [F: b > b,X3: b,Y2: b] :
( ( inj_on_b_b @ F @ top_top_set_b )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) ).
% injD
thf(fact_337_inj__on__imageI2,axiom,
! [F2: b > b,F: a > b,A: set_a] :
( ( inj_on_a_b @ ( comp_b_b_a @ F2 @ F ) @ A )
=> ( inj_on_a_b @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_338_inj__on__imageI2,axiom,
! [F2: c > b,F: a > c,A: set_a] :
( ( inj_on_a_b @ ( comp_c_b_a @ F2 @ F ) @ A )
=> ( inj_on_a_c @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_339_inj__on__imageI2,axiom,
! [F2: a > b,F: a > a,A: set_a] :
( ( inj_on_a_b @ ( comp_a_b_a @ F2 @ F ) @ A )
=> ( inj_on_a_a @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_340_inj__on__imageI2,axiom,
! [F2: b > b,F: c > b,A: set_c] :
( ( inj_on_c_b @ ( comp_b_b_c @ F2 @ F ) @ A )
=> ( inj_on_c_b @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_341_inj__on__imageI2,axiom,
! [F2: a > b,F: c > a,A: set_c] :
( ( inj_on_c_b @ ( comp_a_b_c @ F2 @ F ) @ A )
=> ( inj_on_c_a @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_342_inj__on__imageI2,axiom,
! [F2: b > a,F: c > b,A: set_c] :
( ( inj_on_c_a @ ( comp_b_a_c @ F2 @ F ) @ A )
=> ( inj_on_c_b @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_343_inj__on__imageI2,axiom,
! [F2: a > a,F: c > a,A: set_c] :
( ( inj_on_c_a @ ( comp_a_a_c @ F2 @ F ) @ A )
=> ( inj_on_c_a @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_344_inj__on__imageI2,axiom,
! [F2: b > b,F: b > b,A: set_b] :
( ( inj_on_b_b @ ( comp_b_b_b @ F2 @ F ) @ A )
=> ( inj_on_b_b @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_345_inj__on__imageI2,axiom,
! [F2: b > c,F: a > b,A: set_a] :
( ( inj_on_a_c @ ( comp_b_c_a @ F2 @ F ) @ A )
=> ( inj_on_a_b @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_346_inj__on__imageI2,axiom,
! [F2: c > c,F: a > c,A: set_a] :
( ( inj_on_a_c @ ( comp_c_c_a @ F2 @ F ) @ A )
=> ( inj_on_a_c @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_347_fun_Oinj__map__strong,axiom,
! [X3: c > a,Xa2: c > a,F: a > b,Fa: a > b] :
( ! [Z2: a,Za: a] :
( ( member_a @ Z2 @ ( image_c_a @ X3 @ top_top_set_c ) )
=> ( ( member_a @ Za @ ( image_c_a @ Xa2 @ top_top_set_c ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( comp_a_b_c @ F @ X3 )
= ( comp_a_b_c @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_348_fun_Oinj__map__strong,axiom,
! [X3: a > c,Xa2: a > c,F: c > b,Fa: c > b] :
( ! [Z2: c,Za: c] :
( ( member_c @ Z2 @ ( image_a_c @ X3 @ top_top_set_a ) )
=> ( ( member_c @ Za @ ( image_a_c @ Xa2 @ top_top_set_a ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( comp_c_b_a @ F @ X3 )
= ( comp_c_b_a @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_349_fun_Oinj__map__strong,axiom,
! [X3: a > b,Xa2: a > b,F: b > b,Fa: b > b] :
( ! [Z2: b,Za: b] :
( ( member_b @ Z2 @ ( image_a_b @ X3 @ top_top_set_a ) )
=> ( ( member_b @ Za @ ( image_a_b @ Xa2 @ top_top_set_a ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( comp_b_b_a @ F @ X3 )
= ( comp_b_b_a @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_350_fun_Oinj__map__strong,axiom,
! [X3: a > a,Xa2: a > a,F: a > b,Fa: a > b] :
( ! [Z2: a,Za: a] :
( ( member_a @ Z2 @ ( image_a_a @ X3 @ top_top_set_a ) )
=> ( ( member_a @ Za @ ( image_a_a @ Xa2 @ top_top_set_a ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( comp_a_b_a @ F @ X3 )
= ( comp_a_b_a @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_351_fun_Omap__cong0,axiom,
! [X3: a > b,F: b > b,G: b > b] :
( ! [Z2: b] :
( ( member_b @ Z2 @ ( image_a_b @ X3 @ top_top_set_a ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( comp_b_b_a @ F @ X3 )
= ( comp_b_b_a @ G @ X3 ) ) ) ).
% fun.map_cong0
thf(fact_352_fun_Omap__cong0,axiom,
! [X3: a > a,F: a > b,G: a > b] :
( ! [Z2: a] :
( ( member_a @ Z2 @ ( image_a_a @ X3 @ top_top_set_a ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( comp_a_b_a @ F @ X3 )
= ( comp_a_b_a @ G @ X3 ) ) ) ).
% fun.map_cong0
% Conjectures (3)
thf(conj_0,hypothesis,
inj_on_a_b @ f @ ( sup_sup_set_a @ ( image_c_a @ g @ top_top_set_c ) @ ( image_c_a @ h @ top_top_set_c ) ) ).
thf(conj_1,hypothesis,
( ( comp_a_b_c @ f @ g )
= ( comp_a_b_c @ f @ h ) ) ).
thf(conj_2,conjecture,
g = h ).
%------------------------------------------------------------------------------