TPTP Problem File: ITP098^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP098^2 : 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 : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 325 ( 86 unt; 32 typ; 0 def)
% Number of atoms : 804 ( 270 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3843 ( 27 ~; 3 |; 32 &;3381 @)
% ( 0 <=>; 400 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 278 ( 278 >; 0 *; 0 +; 0 <<)
% Number of symbols : 30 ( 29 usr; 0 con; 1-6 aty)
% Number of variables : 1246 ( 65 ^;1131 !; 20 ?;1246 :)
% ( 30 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:12.554
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (28)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $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 (256)
thf(fact_0_o__ext,axiom,
! [C: $tType,A: $tType,B: $tType,H: B > A,F: A > C,G: A > C] :
( ! [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ H @ ( top_top @ ( set @ B ) ) ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( comp @ A @ C @ B @ F @ H )
= ( comp @ A @ C @ B @ G @ H ) ) ) ).
% o_ext
thf(fact_1_o__cong,axiom,
! [C: $tType,B: $tType,A: $tType,H: A > B,I: A > B,F: B > C,G: B > C] :
( ( H = I )
=> ( ! [X: B] :
( ( member @ B @ X @ ( image @ A @ B @ I @ ( top_top @ ( set @ A ) ) ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( comp @ B @ C @ A @ F @ H )
= ( comp @ B @ C @ A @ F @ I ) ) ) ) ).
% o_cong
thf(fact_2_o__inj__on,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C,H: A > C] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ C @ B @ A @ F @ H ) )
=> ( ( inj_on @ C @ B @ F @ ( sup_sup @ ( set @ C ) @ ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) ) @ ( image @ A @ C @ H @ ( top_top @ ( set @ A ) ) ) ) )
=> ( G = H ) ) ) ).
% o_inj_on
thf(fact_3_ex__o__conv,axiom,
! [B: $tType,A: $tType,C: $tType,G: A > C,F: B > C] :
( ( ? [H2: A > B] :
( G
= ( comp @ B @ C @ A @ F @ H2 ) ) )
= ( ! [X2: C] :
( ( member @ C @ X2 @ ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) ) )
=> ? [Y: B] :
( X2
= ( F @ Y ) ) ) ) ) ).
% ex_o_conv
thf(fact_4_o__eq__conv,axiom,
! [B: $tType,A: $tType,C: $tType,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_5_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X3 )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_6_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_7_inj__on__Un__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( ( ( image @ A @ B @ F @ A2 )
= ( image @ A @ B @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_8_inj__compose,axiom,
! [A: $tType,B: $tType,C: $tType,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_9_fun_Oinj__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( D > A ) @ ( D > B ) @ ( comp @ A @ B @ D @ F ) @ ( top_top @ ( set @ ( D > A ) ) ) ) ) ).
% fun.inj_map
thf(fact_10_comp__inj__on,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A2: set @ A,G: B > C] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( inj_on @ B @ C @ G @ ( image @ A @ B @ F @ A2 ) )
=> ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_11_inj__on__imageI,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A2: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A2 )
=> ( inj_on @ C @ B @ G @ ( image @ A @ C @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_12_comp__inj__on__iff,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A2: set @ A,F2: B > C] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( inj_on @ B @ C @ F2 @ ( image @ A @ B @ F @ A2 ) )
= ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F2 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_13_range__ex1__eq,axiom,
! [B: $tType,A: $tType,F: A > B,B3: B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ B3 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
= ( ? [X2: A] :
( ( B3
= ( F @ X2 ) )
& ! [Y: A] :
( ( B3
= ( F @ Y ) )
=> ( Y = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_14_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ B @ F @ A2 )
= ( image @ A @ B @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_15_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: A,A2: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) )
= ( member @ A @ A3 @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_16_comp__surj,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: A > C] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ C ) ) )
=> ( ( image @ B @ C @ ( comp @ A @ C @ B @ G @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ C ) ) ) ) ) ).
% comp_surj
thf(fact_17_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ B3 )
= ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_18_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
= ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% sup_left_idem
thf(fact_19_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) )
= ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_20_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ X3 )
= X3 ) ) ).
% sup_idem
thf(fact_21_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_22_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_23_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F3: B > A,G2: C > B,X2: C] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_24_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A,Z: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) )
= ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_25_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A3 @ C2 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_26_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% sup_commute
thf(fact_27_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A4: A,B4: A] : ( sup_sup @ A @ B4 @ A4 ) ) ) ) ).
% sup.commute
thf(fact_28_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
= ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) ) ) ) ).
% sup_assoc
thf(fact_29_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ C2 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_30_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( sup_sup @ A @ K @ B3 ) )
=> ( ( sup_sup @ A @ A3 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_31_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,K: A,A3: A,B3: A] :
( ( A2
= ( sup_sup @ A @ K @ A3 ) )
=> ( ( sup_sup @ A @ A2 @ B3 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_32_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_33_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_34_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y2: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
= ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_35_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y2: A,Z: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) )
= ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_36_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
= ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% inf_sup_aci(8)
thf(fact_37_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_38_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B3: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= C2 )
=> ( ( A3 @ ( B3 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_39_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C2: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ! [V2: A] :
( ( A3 @ ( B3 @ V2 ) )
= ( C2 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_40_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C2: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ( ( A3 @ ( B3 @ V ) )
= ( C2 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_41_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% comp_assoc
thf(fact_42_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F3: B > C,G2: A > B,X2: A] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_43_inj__on__inverseI,axiom,
! [B: $tType,A: $tType,A2: set @ A,G: B > A,F: A > B] :
( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on @ A @ B @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_44_inj__on__contraD,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( X3 != Y2 )
=> ( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y2 @ A2 )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_inj__on__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y2 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_50_inj__on__cong,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B] :
( ! [A5: A] :
( ( member @ A @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on @ A @ B @ F @ A2 )
= ( inj_on @ A @ B @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_51_inj__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F3: A > B,A6: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ! [Y: A] :
( ( member @ A @ Y @ A6 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y ) )
=> ( X2 = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_52_inj__onI,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B] :
( ! [X: A,Y3: A] :
( ( member @ A @ X @ A2 )
=> ( ( member @ A @ Y3 @ A2 )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) )
=> ( inj_on @ A @ B @ F @ A2 ) ) ).
% inj_onI
thf(fact_53_inj__onD,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y2 @ A2 )
=> ( X3 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_54_surj__def,axiom,
! [B: $tType,A: $tType,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_55_surjI,axiom,
! [B: $tType,A: $tType,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_56_surjE,axiom,
! [A: $tType,B: $tType,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_57_surjD,axiom,
! [A: $tType,B: $tType,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_58_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A2: set @ B,G: C > A,B2: set @ C,H: A > D] :
( ( ( image @ B @ A @ F @ A2 )
= ( image @ C @ A @ G @ B2 ) )
=> ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A2 )
= ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_59_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,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_60_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A2: set @ A,G: A > B,F: A > A] :
( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ! [Xa: A] :
( ( member @ A @ Xa @ A2 )
=> ( ( ( G @ ( F @ X ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on @ A @ A @ F @ A2 )
=> ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A2 ) )
= ( inj_on @ A @ B @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_61_inj__def,axiom,
! [B: $tType,A: $tType,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_62_inj__eq,axiom,
! [B: $tType,A: $tType,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_63_injI,axiom,
! [B: $tType,A: $tType,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_64_injD,axiom,
! [B: $tType,A: $tType,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_65_inj__on__imageI2,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,F: A > C,A2: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F2 @ F ) @ A2 )
=> ( inj_on @ A @ C @ F @ A2 ) ) ).
% inj_on_imageI2
thf(fact_66_fun_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X3: D > A,Xa2: D > A,F: A > B,Fa: A > B] :
( ! [Z2: A,Za: A] :
( ( member @ A @ Z2 @ ( image @ D @ A @ X3 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ A @ Za @ ( image @ D @ A @ Xa2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( comp @ A @ B @ D @ F @ X3 )
= ( comp @ A @ B @ D @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_67_fun_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X3: D > A,F: A > B,G: A > B] :
( ! [Z2: A] :
( ( member @ A @ Z2 @ ( image @ D @ A @ X3 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( comp @ A @ B @ D @ F @ X3 )
= ( comp @ A @ B @ D @ G @ X3 ) ) ) ).
% fun.map_cong0
thf(fact_68_fun_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X3: D > A,Ya: D > A,F: A > B,G: A > B] :
( ( X3 = Ya )
=> ( ! [Z2: A] :
( ( member @ A @ Z2 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( comp @ A @ B @ D @ F @ X3 )
= ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_69_fun_Oset__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: A > B,V: D > A] :
( ( image @ D @ B @ ( comp @ A @ B @ D @ F @ V ) @ ( top_top @ ( set @ D ) ) )
= ( image @ A @ B @ F @ ( image @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% fun.set_map
thf(fact_70_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > A,X4: set @ B,G1: A > C,G22: A > C] :
( ( ( image @ B @ A @ F @ X4 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ X4 )
=> ( ( comp @ A @ C @ B @ G1 @ F @ X )
= ( comp @ A @ C @ B @ G22 @ F @ X ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_71_Un__iff,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C2 @ A2 )
| ( member @ A @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_72_UnCI,axiom,
! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ A2 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_73_iso__tuple__UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_74_UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_75_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_76_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X3: B,A2: set @ B] :
( ( B3
= ( F @ X3 ) )
=> ( ( member @ B @ X3 @ A2 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_77_inj__imp__inj__on,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ B @ F @ A2 ) ) ).
% inj_imp_inj_on
thf(fact_78_Un__UNIV__right,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_79_Un__UNIV__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_80_imageI,axiom,
! [B: $tType,A: $tType,X3: A,A2: set @ A,F: A > B] :
( ( member @ A @ X3 @ A2 )
=> ( member @ B @ ( F @ X3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% imageI
thf(fact_81_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F: B > A,A2: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F @ A2 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_82_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X: B] :
( ( member @ B @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_83_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
( ( M = N )
=> ( ! [X: A] :
( ( member @ A @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N ) ) ) ) ).
% image_cong
thf(fact_84_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_85_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X3: A,A2: set @ A,B3: B,F: A > B] :
( ( member @ A @ X3 @ A2 )
=> ( ( B3
= ( F @ X3 ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_86_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X: A] : ( member @ A @ X @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_87_UNIV__witness,axiom,
! [A: $tType] :
? [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_88_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_89_UnE,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( ~ ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_90_UnI1,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_91_UnI2,axiom,
! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_92_bex__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_93_ball__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_94_Un__assoc,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_assoc
thf(fact_95_Un__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_96_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% Un_commute
thf(fact_97_Un__left__absorb,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_98_Un__left__commute,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_99_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X3: B] : ( member @ A @ ( F @ X3 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_100_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X3: B] :
( ( B3
= ( F @ X3 ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_101_image__Un,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ B] :
( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ ( image @ B @ A @ F @ B2 ) ) ) ).
% image_Un
thf(fact_102_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A2: set @ C] :
( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A2 ) ) )
= ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_103_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A2: set @ C] :
( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A2 ) ) )
= ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_104_the__inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A2: set @ C,X3: B] :
( ( inj_on @ A @ B @ F @ ( image @ C @ A @ G @ A2 ) )
=> ( ( inj_on @ C @ A @ G @ A2 )
=> ( ( member @ B @ X3 @ ( image @ A @ B @ F @ ( image @ C @ A @ G @ A2 ) ) )
=> ( ( the_inv_into @ C @ B @ A2 @ ( comp @ A @ B @ C @ F @ G ) @ X3 )
= ( comp @ A @ C @ B @ ( the_inv_into @ C @ A @ A2 @ G ) @ ( the_inv_into @ A @ B @ ( image @ C @ A @ G @ A2 ) @ F ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_105_the__inv__into__onto,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_106_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X3: C,F2: D > A,G3: E > D,X6: E] :
( ( ( F @ ( G @ X3 ) )
= ( F2 @ ( G3 @ X6 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X3 )
= ( comp @ D @ A @ E @ F2 @ G3 @ X6 ) ) ) ).
% comp_cong
thf(fact_107_comp2__conv,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comp @ C @ B @ A )
= ( ^ [F1: C > B,F22: A > C,X2: A] : ( F1 @ ( F22 @ X2 ) ) ) ) ).
% comp2_conv
thf(fact_108_top1I,axiom,
! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% top1I
thf(fact_109_the__inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: B] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ( F @ X3 )
= Y2 )
=> ( ( member @ A @ X3 @ A2 )
=> ( ( the_inv_into @ A @ B @ A2 @ F @ Y2 )
= X3 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_110_the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( member @ A @ X3 @ A2 )
=> ( ( the_inv_into @ A @ B @ A2 @ F @ ( F @ X3 ) )
= X3 ) ) ) ).
% the_inv_into_f_f
thf(fact_111_inj__on__the__inv__into,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_112_f__the__inv__into__f,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ A,Y2: B] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( member @ B @ Y2 @ ( image @ A @ B @ F @ A2 ) )
=> ( ( F @ ( the_inv_into @ A @ B @ A2 @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_113_the__inv__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,X3: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X3 ) )
= X3 ) ) ).
% the_inv_f_f
thf(fact_114_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X: B] :
( ( member @ B @ X @ B2 )
=> ( ( C3 @ X )
= ( D3 @ X ) ) )
=> ( ( Sup @ ( image @ B @ A @ C3 @ A2 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_115_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X: B] :
( ( member @ B @ X @ B2 )
=> ( ( C3 @ X )
= ( D3 @ X ) ) )
=> ( ( Inf @ ( image @ B @ A @ C3 @ A2 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_116_comp3__conv,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F12: D > B,F23: C > D,F32: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F12 @ F23 ) @ F32 )
= ( ^ [X2: A] : ( F12 @ ( F23 @ ( F32 @ X2 ) ) ) ) ) ).
% comp3_conv
thf(fact_117_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_118_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X3: A] :
( ( P
& ( top_top @ ( A > $o ) @ X3 ) )
= P ) ).
% top_conj(2)
thf(fact_119_top__conj_I1_J,axiom,
! [A: $tType,X3: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X3 )
& P )
= P ) ).
% top_conj(1)
thf(fact_120_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: B,B2: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( member @ B @ X3 @ ( image @ A @ B @ F @ A2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A2 @ F @ X3 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_121_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R2: A > D,F: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G @ H )
= ( comp @ D @ B @ A @ R1 @ R2 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R1 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
= ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_122_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_123_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_124_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_125_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
= ( ( ord_less_eq @ A @ X3 @ Z )
& ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_126_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
= ( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_127_Un__subset__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_128_bex__cong2,axiom,
! [B: $tType,A: $tType,I2: set @ A,A2: set @ A,F: A > B,G: A > B,P: B > $o] :
( ( ord_less_eq @ ( set @ A ) @ I2 @ A2 )
=> ( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( ? [X2: A] :
( ( member @ A @ X2 @ I2 )
& ( P @ ( F @ X2 ) ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ I2 )
& ( P @ ( G @ X2 ) ) ) ) ) ) ) ).
% bex_cong2
thf(fact_129_ball__cong2,axiom,
! [B: $tType,A: $tType,I2: set @ A,A2: set @ A,F: A > B,G: A > B,P: B > $o] :
( ( ord_less_eq @ ( set @ A ) @ I2 @ A2 )
=> ( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( ! [X2: A] :
( ( member @ A @ X2 @ I2 )
=> ( P @ ( F @ X2 ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ I2 )
=> ( P @ ( G @ X2 ) ) ) ) ) ) ) ).
% ball_cong2
thf(fact_130_bex__subset__imp__bex,axiom,
! [A: $tType,A2: set @ A,P: A > $o,B2: set @ A] :
( ? [X5: A] :
( ( member @ A @ X5 @ A2 )
& ( P @ X5 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ? [X: A] :
( ( member @ A @ X @ B2 )
& ( P @ X ) ) ) ) ).
% bex_subset_imp_bex
thf(fact_131_ball__subset__imp__ball,axiom,
! [A: $tType,B2: set @ A,P: A > $o,A2: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ B2 )
=> ( P @ X ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ! [X5: A] :
( ( member @ A @ X5 @ A2 )
=> ( P @ X5 ) ) ) ) ).
% ball_subset_imp_ball
thf(fact_132_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_133_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_134_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_135_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
& ( B2
= ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_136_image__subset__iff,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A2 )
=> ( member @ A @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_137_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
=> ~ ! [C4: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C4 @ A2 )
=> ( B2
!= ( image @ B @ A @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_138_image__subsetI,axiom,
! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( member @ B @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_139_image__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% image_mono
thf(fact_140_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_141_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.coboundedI2
thf(fact_142_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.coboundedI1
thf(fact_143_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( ( sup_sup @ A @ A4 @ B4 )
= B4 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_144_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( ( sup_sup @ A @ A4 @ B4 )
= A4 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_145_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.cobounded2
thf(fact_146_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.cobounded1
thf(fact_147_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( A4
= ( sup_sup @ A @ A4 @ B4 ) ) ) ) ) ).
% sup.order_iff
thf(fact_148_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_149_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A3 )
=> ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_150_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( sup_sup @ A @ X3 @ Y2 )
= Y2 ) ) ) ).
% sup_absorb2
thf(fact_151_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ( sup_sup @ A @ X3 @ Y2 )
= X3 ) ) ) ).
% sup_absorb1
thf(fact_152_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb2
thf(fact_153_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_154_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F: A > A > A,X3: A,Y2: A] :
( ! [X: A,Y3: A] : ( ord_less_eq @ A @ X @ ( F @ X @ Y3 ) )
=> ( ! [X: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F @ X @ Y3 ) )
=> ( ! [X: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( ord_less_eq @ A @ Z2 @ X )
=> ( ord_less_eq @ A @ ( F @ Y3 @ Z2 ) @ X ) ) )
=> ( ( sup_sup @ A @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ) ).
% sup_unique
thf(fact_155_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% sup.orderI
thf(fact_156_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.orderE
thf(fact_157_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y: A] :
( ( sup_sup @ A @ X2 @ Y )
= Y ) ) ) ) ).
% le_iff_sup
thf(fact_158_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y2: A,X3: A,Z: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ( ord_less_eq @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y2 @ Z ) @ X3 ) ) ) ) ).
% sup_least
thf(fact_159_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,C2: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_160_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,D2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ A @ D2 @ B3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% sup.mono
thf(fact_161_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ X3 @ B3 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% le_supI2
thf(fact_162_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X3 @ A3 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% le_supI1
thf(fact_163_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y2: A,X3: A] : ( ord_less_eq @ A @ Y2 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% sup_ge2
thf(fact_164_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% sup_ge1
thf(fact_165_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
=> ( ( ord_less_eq @ A @ B3 @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X3 ) ) ) ) ).
% le_supI
thf(fact_166_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A,X3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X3 )
=> ~ ( ( ord_less_eq @ A @ A3 @ X3 )
=> ~ ( ord_less_eq @ A @ B3 @ X3 ) ) ) ) ).
% le_supE
thf(fact_167_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y2: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% inf_sup_ord(3)
thf(fact_168_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y2: A,X3: A] : ( ord_less_eq @ A @ Y2 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% inf_sup_ord(4)
thf(fact_169_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_170_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_171_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_172_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( member @ A @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_173_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_174_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_175_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [T: A] :
( ( member @ A @ T @ A6 )
=> ( member @ A @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_176_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_177_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_178_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_179_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_180_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_181_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_182_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z3: set @ A] : Y4 = Z3 )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_183_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_184_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_185_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X: B,Y3: B] :
( ( ord_less_eq @ B @ X @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_186_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_187_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X: B,Y3: B] :
( ( ord_less_eq @ B @ X @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_188_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_189_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
& ( ord_less_eq @ A @ Y @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_190_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ) ).
% antisym
thf(fact_191_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% linear
thf(fact_192_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( X3 = Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% eq_refl
thf(fact_193_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% le_cases
thf(fact_194_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_195_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A,Z: A] :
( ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_196_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_197_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_198_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_199_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_200_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_201_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% order_trans
thf(fact_202_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_203_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B6: A] :
( ( ord_less_eq @ A @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: A,B6: A] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_204_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_205_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_206_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_207_Un__mono,axiom,
! [A: $tType,A2: set @ A,C3: set @ A,B2: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_208_Un__least,axiom,
! [A: $tType,A2: set @ A,C3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 ) ) ) ).
% Un_least
thf(fact_209_Un__upper1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_210_Un__upper2,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_211_Un__absorb1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_212_Un__absorb2,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_213_subset__UnE,axiom,
! [A: $tType,C3: set @ A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ A2 )
=> ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ B2 )
=> ( C3
!= ( sup_sup @ ( set @ A ) @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_214_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_215_inj__on__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( inj_on @ A @ B @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_216_subset__inj__on,axiom,
! [B: $tType,A: $tType,F: A > B,B2: set @ A,A2: set @ A] :
( ( inj_on @ A @ B @ F @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_217_SetInterval2_Oinj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A,A3: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ B2 ) )
= ( member @ A @ A3 @ B2 ) ) ) ) ) ).
% SetInterval2.inj_on_image_mem_iff
thf(fact_218_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,C3: set @ A,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ( ( image @ A @ B @ F @ A2 )
= ( image @ A @ B @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_219_Fun_Oinj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,B2: set @ A,A3: A,A2: set @ A] :
( ( inj_on @ A @ B @ F @ B2 )
=> ( ( member @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) )
= ( member @ A @ A3 @ A2 ) ) ) ) ) ).
% Fun.inj_on_image_mem_iff
thf(fact_220_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_221_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F @ G )
= L )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
= ( comp @ A @ B @ D @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_222_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G @ H )
= R )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
= ( comp @ B @ D @ A @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_223_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H )
= R )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
= ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_224_subset__image__inj,axiom,
! [A: $tType,B: $tType,S: set @ A,F: B > A,T2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( image @ B @ A @ F @ T2 ) )
= ( ? [U: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U @ T2 )
& ( inj_on @ B @ A @ F @ U )
& ( S
= ( image @ B @ A @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_225_ex__subset__image__inj,axiom,
! [A: $tType,B: $tType,F: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ? [T3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T3 @ ( image @ B @ A @ F @ S ) )
& ( P @ T3 ) ) )
= ( ? [T3: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ T3 @ S )
& ( inj_on @ B @ A @ F @ T3 )
& ( P @ ( image @ B @ A @ F @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_226_all__subset__image__inj,axiom,
! [A: $tType,B: $tType,F: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ! [T3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T3 @ ( image @ B @ A @ F @ S ) )
=> ( P @ T3 ) ) )
= ( ! [T3: set @ B] :
( ( ( ord_less_eq @ ( set @ B ) @ T3 @ S )
& ( inj_on @ B @ A @ F @ T3 ) )
=> ( P @ ( image @ B @ A @ F @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_227_all__subset__image,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: ( set @ A ) > $o] :
( ( ! [B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A2 )
=> ( P @ ( image @ B @ A @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_228_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_229_inj__on__image__Fpow,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image @ A @ B @ F ) @ ( finite_Fpow @ A @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_230_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X2: A] : ( uminus_uminus @ B @ ( A6 @ X2 ) ) ) ) ) ).
% uminus_apply
thf(fact_231_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X3 ) )
= X3 ) ) ).
% double_compl
thf(fact_232_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A,Y2: A] :
( ( ( uminus_uminus @ A @ X3 )
= ( uminus_uminus @ A @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% compl_eq_compl_iff
thf(fact_233_ComplI,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ~ ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% ComplI
thf(fact_234_Compl__iff,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_235_Compl__eq__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A2 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( A2 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_236_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% compl_le_compl_iff
thf(fact_237_inj__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A2 ) ) ).
% inj_uminus
thf(fact_238_Compl__subset__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_239_Compl__anti__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_240_sup__compl__top__left2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A,Y2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left2
thf(fact_241_sup__compl__top__left1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A,Y2: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ ( sup_sup @ A @ X3 @ Y2 ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left1
thf(fact_242_sup__compl__top,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( uminus_uminus @ A @ X3 ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top
thf(fact_243_compl__sup__top,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ X3 )
= ( top_top @ A ) ) ) ).
% compl_sup_top
thf(fact_244_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% compl_mono
thf(fact_245_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ ( uminus_uminus @ A @ X3 ) )
=> ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% compl_le_swap1
thf(fact_246_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ X3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) ) ) ).
% compl_le_swap2
thf(fact_247_Fpow__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% Fpow_mono
thf(fact_248_sup__cancel__left2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A,A3: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ A3 ) @ ( sup_sup @ A @ X3 @ B3 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left2
thf(fact_249_sup__cancel__left1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A,A3: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ B3 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left1
thf(fact_250_Compl__partition2,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ A2 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_251_Compl__partition,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_252_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X2: A] : ( uminus_uminus @ B @ ( A6 @ X2 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_253_ComplD,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
=> ~ ( member @ A @ C2 @ A2 ) ) ).
% ComplD
thf(fact_254_double__complement,axiom,
! [A: $tType,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_255_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) ) @ ( image @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A2 ) ) ) ) ).
% surj_Compl_image_subset
% Type constructors (34)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice_top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 )
=> ( semilattice_sup @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A8: $tType,A9: $tType] :
( ( boolean_algebra @ A9 )
=> ( boolean_algebra @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A8: $tType,A9: $tType] :
( ( order_top @ A9 )
=> ( order_top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 )
=> ( lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 )
=> ( top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A8: $tType,A9: $tType] :
( ( uminus @ A9 )
=> ( uminus @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_3,axiom,
! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_4,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_5,axiom,
! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_6,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_10,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_11,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_12,axiom,
! [A8: $tType] : ( uminus @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_13,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_15,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_16,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_17,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_18,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_19,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_20,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_21,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_22,axiom,
uminus @ $o ).
% 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 ).
%------------------------------------------------------------------------------