TPTP Problem File: CAT036+1.p
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%------------------------------------------------------------------------------
% File : CAT036+1 : TPTP v9.0.0. Released v3.4.0.
% Domain : Category Theory
% Problem : Yoneda Embedding T08
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Woj97] Wojciechowski (1997), Yoneda Embedding
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t8_yoneda_1 [Urb08]
% Status : Unknown
% Rating : 1.00 v3.4.0
% Syntax : Number of formulae : 137 ( 28 unt; 0 def)
% Number of atoms : 652 ( 72 equ)
% Maximal formula atoms : 32 ( 4 avg)
% Number of connectives : 559 ( 44 ~; 2 |; 318 &)
% ( 13 <=>; 182 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 25 ( 23 usr; 1 prp; 0-6 aty)
% Number of functors : 50 ( 50 usr; 1 con; 0-7 aty)
% Number of variables : 357 ( 341 !; 16 ?)
% SPC : FOF_UNK_RFO_SEQ
% Comments : Normal version: includes the axioms (which may be theorems from
% other articles) and background that are possibly necessary.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
% : Infinox says this has no finite (counter-) models.
%------------------------------------------------------------------------------
fof(t8_yoneda_1,conjecture,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> v2_funct_1(k4_yoneda_1(A)) ) ).
fof(abstractness_v1_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ( v1_cat_1(A)
=> A = g1_cat_1(u1_cat_1(A),u2_cat_1(A),u3_cat_1(A),u4_cat_1(A),u5_cat_1(A),u6_cat_1(A)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_funct_1(A) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> v1_relat_1(C) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) ) ) ).
fof(commutativity_k2_tarski,axiom,
! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ).
fof(d18_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v1_cat_1(C)
& v2_cat_1(C)
& l1_cat_1(C) )
=> ( C = k12_nattra_1(A,B)
<=> ( u1_cat_1(C) = k7_cat_2(A,B)
& u2_cat_1(C) = k11_nattra_1(A,B)
& ! [D] :
( m1_subset_1(D,u2_cat_1(C))
=> ( k2_cat_1(C,D) = k1_mcart_1(k1_mcart_1(D))
& k3_cat_1(C,D) = k2_mcart_1(k1_mcart_1(D)) ) )
& ! [D] :
( m1_subset_1(D,u2_cat_1(C))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(C))
=> ( k2_cat_1(C,E) = k3_cat_1(C,D)
=> r2_hidden(k13_cat_2(C,C,E,D),k1_relat_1(u5_cat_1(C))) ) ) )
& ! [D] :
( m1_subset_1(D,u2_cat_1(C))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(C))
=> ~ ( r2_hidden(k13_cat_2(C,C,E,D),k1_relat_1(u5_cat_1(C)))
& ! [F] :
( m2_cat_1(F,A,B)
=> ! [G] :
( m2_cat_1(G,A,B)
=> ! [H] :
( m2_cat_1(H,A,B)
=> ! [I] :
( m2_nattra_1(I,A,B,F,G)
=> ! [J] :
( m2_nattra_1(J,A,B,G,H)
=> ~ ( D = k4_tarski(k4_tarski(F,G),I)
& E = k4_tarski(k4_tarski(G,H),J)
& k1_funct_1(u5_cat_1(C),k13_cat_2(C,C,E,D)) = k4_tarski(k4_tarski(F,H),k8_nattra_1(A,B,F,G,H,I,J)) ) ) ) ) ) ) ) ) )
& ! [D] :
( m1_subset_1(D,u1_cat_1(C))
=> ! [E] :
( m2_cat_1(E,A,B)
=> ( E = D
=> k10_cat_1(C,D) = k4_tarski(k4_tarski(E,E),k7_nattra_1(A,B,E)) ) ) ) ) ) ) ) ) ).
fof(d1_funct_2,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ( ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
=> ( v1_funct_2(C,A,B)
<=> A = k4_relset_1(A,B,C) ) )
& ( B = k1_xboole_0
=> ( A = k1_xboole_0
| ( v1_funct_2(C,A,B)
<=> C = k1_xboole_0 ) ) ) ) ) ).
fof(d1_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ( k6_cat_1(A,D,E) != k1_xboole_0
=> ! [F] :
( m1_cat_1(F,A,D,E)
=> k3_nattra_1(A,B,C,D,E,F) = k8_funct_2(u2_cat_1(A),u2_cat_1(B),C,F) ) ) ) ) ) ) ) ).
fof(d1_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> k1_yoneda_1(A) = k12_ens_1(k17_ens_1(A)) ) ).
fof(d22_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u2_cat_1(A),k2_ens_1(k17_ens_1(A)))
& m2_relset_1(C,u2_cat_1(A),k2_ens_1(k17_ens_1(A))) )
=> ( C = k20_ens_1(A,B)
<=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> k8_funct_2(u2_cat_1(A),k2_ens_1(k17_ens_1(A)),C,D) = k1_domain_1(k2_zfmisc_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A))),k1_zfmisc_1(k2_zfmisc_1(k6_cat_1(A,B,k2_cat_1(A,D)),k6_cat_1(A,B,k3_cat_1(A,D)))),k1_domain_1(k1_zfmisc_1(u2_cat_1(A)),k1_zfmisc_1(u2_cat_1(A)),k6_cat_1(A,B,k2_cat_1(A,D)),k6_cat_1(A,B,k3_cat_1(A,D))),k18_ens_1(A,B,D)) ) ) ) ) ) ).
fof(d2_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( C = k5_cat_2(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> m2_cat_1(D,A,B) ) ) ) ) ).
fof(d2_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> k2_yoneda_1(A,B) = k20_ens_1(A,B) ) ) ).
fof(d3_mcart_1,axiom,
! [A,B,C] : k3_mcart_1(A,B,C) = k4_tarski(k4_tarski(A,B),C) ).
fof(d3_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m2_nattra_1(C,A,k1_yoneda_1(A),k2_yoneda_1(A,k3_cat_1(A,B)),k2_yoneda_1(A,k2_cat_1(A,B)))
=> ( C = k3_yoneda_1(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> k5_nattra_1(A,k1_yoneda_1(A),k2_yoneda_1(A,k3_cat_1(A,B)),k2_yoneda_1(A,k2_cat_1(A,B)),C,D) = k4_tarski(k4_tarski(k6_cat_1(A,k3_cat_1(A,B),D),k6_cat_1(A,k2_cat_1(A,B),D)),k22_ens_1(A,B,k10_cat_1(A,D))) ) ) ) ) ) ).
fof(d4_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_oppcat_1(B,A,k12_nattra_1(A,k1_yoneda_1(A)))
=> ( B = k4_yoneda_1(A)
<=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> k8_funct_2(u2_cat_1(A),u2_cat_1(k12_nattra_1(A,k1_yoneda_1(A))),B,C) = k4_tarski(k4_tarski(k2_yoneda_1(A,k3_cat_1(A,C)),k2_yoneda_1(A,k2_cat_1(A,C))),k3_yoneda_1(A,C)) ) ) ) ) ).
fof(d5_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r1_nattra_1(A,B,C,D)
=> ! [E] :
( m1_nattra_1(E,A,B,C,D)
=> ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> k5_nattra_1(A,B,C,D,E,F) = k8_funct_2(u1_cat_1(A),u2_cat_1(B),E,F) ) ) ) ) ) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ).
fof(d7_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ( k6_cat_1(A,B,C) != k1_xboole_0
=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> ( m1_cat_1(D,A,B,C)
<=> r2_hidden(D,k6_cat_1(A,B,C)) ) ) ) ) ) ) ).
fof(d8_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
<=> ! [B,C] :
( ( r2_hidden(B,k1_relat_1(A))
& r2_hidden(C,k1_relat_1(A))
& k1_funct_1(A,B) = k1_funct_1(A,C) )
=> B = C ) ) ) ).
fof(dt_g1_cat_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,B,A)
& m1_relset_1(C,B,A)
& v1_funct_1(D)
& v1_funct_2(D,B,A)
& m1_relset_1(D,B,A)
& v1_funct_1(E)
& m1_relset_1(E,k2_zfmisc_1(B,B),B)
& v1_funct_1(F)
& v1_funct_2(F,A,B)
& m1_relset_1(F,A,B) )
=> ( v1_cat_1(g1_cat_1(A,B,C,D,E,F))
& l1_cat_1(g1_cat_1(A,B,C,D,E,F)) ) ) ).
fof(dt_k10_cat_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> m1_cat_1(k10_cat_1(A,B),A,B,B) ) ).
fof(dt_k11_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ( v2_cat_1(k11_cat_2(A,B))
& l1_cat_1(k11_cat_2(A,B)) ) ) ).
fof(dt_k11_nattra_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> m4_nattra_1(k11_nattra_1(A,B),A,B) ) ).
fof(dt_k12_cat_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u2_cat_1(A),u2_cat_1(B))
& m1_relset_1(C,u2_cat_1(A),u2_cat_1(B)) )
=> ( v1_funct_1(k12_cat_1(A,B,C))
& v1_funct_2(k12_cat_1(A,B,C),u1_cat_1(A),u1_cat_1(B))
& m2_relset_1(k12_cat_1(A,B,C),u1_cat_1(A),u1_cat_1(B)) ) ) ).
fof(dt_k12_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> l1_cat_1(k12_ens_1(A)) ) ).
fof(dt_k12_nattra_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ( v1_cat_1(k12_nattra_1(A,B))
& v2_cat_1(k12_nattra_1(A,B))
& l1_cat_1(k12_nattra_1(A,B)) ) ) ).
fof(dt_k13_cat_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m1_subset_1(D,u1_cat_1(A)) )
=> m1_subset_1(k13_cat_1(A,B,C,D),u1_cat_1(B)) ) ).
fof(dt_k13_cat_2,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_subset_1(C,u2_cat_1(A))
& m1_subset_1(D,u2_cat_1(B)) )
=> m1_subset_1(k13_cat_2(A,B,C,D),u2_cat_1(k11_cat_2(A,B))) ) ).
fof(dt_k17_ens_1,axiom,
$true ).
fof(dt_k18_ens_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(A)) )
=> ( v1_funct_1(k18_ens_1(A,B,C))
& v1_funct_2(k18_ens_1(A,B,C),k6_cat_1(A,B,k2_cat_1(A,C)),k6_cat_1(A,B,k3_cat_1(A,C)))
& m2_relset_1(k18_ens_1(A,B,C),k6_cat_1(A,B,k2_cat_1(A,C)),k6_cat_1(A,B,k3_cat_1(A,C))) ) ) ).
fof(dt_k19_ens_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(A)) )
=> ( v1_funct_1(k19_ens_1(A,B,C))
& v1_funct_2(k19_ens_1(A,B,C),k6_cat_1(A,k3_cat_1(A,C),B),k6_cat_1(A,k2_cat_1(A,C),B))
& m2_relset_1(k19_ens_1(A,B,C),k6_cat_1(A,k3_cat_1(A,C),B),k6_cat_1(A,k2_cat_1(A,C),B)) ) ) ).
fof(dt_k1_domain_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,A)
& m1_subset_1(D,B) )
=> m1_subset_1(k1_domain_1(A,B,C,D),k2_zfmisc_1(A,B)) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_mcart_1,axiom,
$true ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ( v2_cat_1(k1_yoneda_1(A))
& l1_cat_1(k1_yoneda_1(A)) ) ) ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k20_ens_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> ( v1_funct_1(k20_ens_1(A,B))
& v1_funct_2(k20_ens_1(A,B),u2_cat_1(A),k2_ens_1(k17_ens_1(A)))
& m2_relset_1(k20_ens_1(A,B),u2_cat_1(A),k2_ens_1(k17_ens_1(A))) ) ) ).
fof(dt_k22_ens_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A))
& m1_subset_1(C,u2_cat_1(A)) )
=> ( v1_funct_1(k22_ens_1(A,B,C))
& v1_funct_2(k22_ens_1(A,B,C),k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,C)),k6_cat_1(A,k2_cat_1(A,B),k3_cat_1(A,C)))
& m2_relset_1(k22_ens_1(A,B,C),k6_cat_1(A,k3_cat_1(A,B),k2_cat_1(A,C)),k6_cat_1(A,k2_cat_1(A,B),k3_cat_1(A,C))) ) ) ).
fof(dt_k2_cat_1,axiom,
! [A,B] :
( ( l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> m1_subset_1(k2_cat_1(A,B),u1_cat_1(A)) ) ).
fof(dt_k2_ens_1,axiom,
$true ).
fof(dt_k2_mcart_1,axiom,
$true ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k2_yoneda_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> m2_cat_1(k2_yoneda_1(A,B),A,k1_yoneda_1(A)) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_cat_1,axiom,
! [A,B] :
( ( l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> m1_subset_1(k3_cat_1(A,B),u1_cat_1(A)) ) ).
fof(dt_k3_mcart_1,axiom,
$true ).
fof(dt_k3_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m1_subset_1(D,u1_cat_1(A))
& m1_subset_1(E,u1_cat_1(A))
& m1_cat_1(F,A,D,E) )
=> m1_cat_1(k3_nattra_1(A,B,C,D,E,F),B,k13_cat_1(A,B,C,D),k13_cat_1(A,B,C,E)) ) ).
fof(dt_k3_yoneda_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> m2_nattra_1(k3_yoneda_1(A,B),A,k1_yoneda_1(A),k2_yoneda_1(A,k3_cat_1(A,B)),k2_yoneda_1(A,k2_cat_1(A,B))) ) ).
fof(dt_k4_nattra_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B) )
=> m1_nattra_1(k4_nattra_1(A,B,C),A,B,C,C) ) ).
fof(dt_k4_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(dt_k4_tarski,axiom,
$true ).
fof(dt_k4_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> m1_oppcat_1(k4_yoneda_1(A),A,k12_nattra_1(A,k1_yoneda_1(A))) ) ).
fof(dt_k5_cat_1,axiom,
! [A,B] :
( ( l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> m1_subset_1(k5_cat_1(A,B),u2_cat_1(A)) ) ).
fof(dt_k5_cat_2,axiom,
$true ).
fof(dt_k5_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B)
& m1_nattra_1(E,A,B,C,D)
& m1_subset_1(F,u1_cat_1(A)) )
=> m1_cat_1(k5_nattra_1(A,B,C,D,E,F),B,k13_cat_1(A,B,C,F),k13_cat_1(A,B,D,F)) ) ).
fof(dt_k6_cat_1,axiom,
! [A,B,C] :
( ( l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u1_cat_1(A)) )
=> m1_subset_1(k6_cat_1(A,B,C),k1_zfmisc_1(u2_cat_1(A))) ) ).
fof(dt_k7_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> m1_cat_2(k7_cat_2(A,B),A,B) ) ).
fof(dt_k7_nattra_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B) )
=> m2_nattra_1(k7_nattra_1(A,B,C),A,B,C,C) ) ).
fof(dt_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,A) )
=> m1_subset_1(k8_funct_2(A,B,C,D),B) ) ).
fof(dt_k8_nattra_1,axiom,
! [A,B,C,D,E,F,G] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B)
& m2_cat_1(E,A,B)
& m2_nattra_1(F,A,B,C,D)
& m2_nattra_1(G,A,B,D,E) )
=> m2_nattra_1(k8_nattra_1(A,B,C,D,E,F,G),A,B,C,E) ) ).
fof(dt_l1_cat_1,axiom,
$true ).
fof(dt_m1_cat_1,axiom,
! [A,B,C] :
( ( l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u1_cat_1(A)) )
=> ! [D] :
( m1_cat_1(D,A,B,C)
=> m1_subset_1(D,u2_cat_1(A)) ) ) ).
fof(dt_m1_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_cat_2(C,A,B)
=> ~ v1_xboole_0(C) ) ) ).
fof(dt_m1_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> ! [E] :
( m1_nattra_1(E,A,B,C,D)
=> ( v1_funct_1(E)
& v1_funct_2(E,u1_cat_1(A),u2_cat_1(B))
& m2_relset_1(E,u1_cat_1(A),u2_cat_1(B)) ) ) ) ).
fof(dt_m1_oppcat_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_oppcat_1(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,u2_cat_1(A),u2_cat_1(B))
& m2_relset_1(C,u2_cat_1(A),u2_cat_1(B)) ) ) ) ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_cat_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,u2_cat_1(A),u2_cat_1(B))
& m2_relset_1(C,u2_cat_1(A),u2_cat_1(B)) ) ) ) ).
fof(dt_m2_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> ! [E] :
( m2_nattra_1(E,A,B,C,D)
=> m1_nattra_1(E,A,B,C,D) ) ) ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ).
fof(dt_m4_nattra_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m4_nattra_1(C,A,B)
=> ~ v1_xboole_0(C) ) ) ).
fof(dt_u1_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ~ v1_xboole_0(u1_cat_1(A)) ) ).
fof(dt_u2_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ~ v1_xboole_0(u2_cat_1(A)) ) ).
fof(dt_u3_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ( v1_funct_1(u3_cat_1(A))
& v1_funct_2(u3_cat_1(A),u2_cat_1(A),u1_cat_1(A))
& m2_relset_1(u3_cat_1(A),u2_cat_1(A),u1_cat_1(A)) ) ) ).
fof(dt_u4_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ( v1_funct_1(u4_cat_1(A))
& v1_funct_2(u4_cat_1(A),u2_cat_1(A),u1_cat_1(A))
& m2_relset_1(u4_cat_1(A),u2_cat_1(A),u1_cat_1(A)) ) ) ).
fof(dt_u5_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ( v1_funct_1(u5_cat_1(A))
& m2_relset_1(u5_cat_1(A),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(A)),u2_cat_1(A)) ) ) ).
fof(dt_u6_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ( v1_funct_1(u6_cat_1(A))
& v1_funct_2(u6_cat_1(A),u1_cat_1(A),u2_cat_1(A))
& m2_relset_1(u6_cat_1(A),u1_cat_1(A),u2_cat_1(A)) ) ) ).
fof(existence_l1_cat_1,axiom,
? [A] : l1_cat_1(A) ).
fof(existence_m1_cat_1,axiom,
! [A,B,C] :
( ( l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u1_cat_1(A)) )
=> ? [D] : m1_cat_1(D,A,B,C) ) ).
fof(existence_m1_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ? [C] : m1_cat_2(C,A,B) ) ).
fof(existence_m1_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> ? [E] : m1_nattra_1(E,A,B,C,D) ) ).
fof(existence_m1_oppcat_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ? [C] : m1_oppcat_1(C,A,B) ) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : m1_relset_1(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(existence_m2_cat_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ? [C] : m2_cat_1(C,A,B) ) ).
fof(existence_m2_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> ? [E] : m2_nattra_1(E,A,B,C,D) ) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
fof(existence_m4_nattra_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ? [C] : m4_nattra_1(C,A,B) ) ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
fof(fc2_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ v1_xboole_0(k2_ens_1(A)) ) ).
fof(fc3_ens_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k2_ens_1(A)) )
=> ( v1_relat_1(k2_mcart_1(B))
& v1_funct_1(k2_mcart_1(B)) ) ) ).
fof(fc4_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_cat_1(k12_ens_1(A))
& v2_cat_1(k12_ens_1(A)) ) ) ).
fof(fc5_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ~ v1_xboole_0(k17_ens_1(A)) ) ).
fof(free_g1_cat_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,B,A)
& m1_relset_1(C,B,A)
& v1_funct_1(D)
& v1_funct_2(D,B,A)
& m1_relset_1(D,B,A)
& v1_funct_1(E)
& m1_relset_1(E,k2_zfmisc_1(B,B),B)
& v1_funct_1(F)
& v1_funct_2(F,A,B)
& m1_relset_1(F,A,B) )
=> ! [G,H,I,J,K,L] :
( g1_cat_1(A,B,C,D,E,F) = g1_cat_1(G,H,I,J,K,L)
=> ( A = G
& B = H
& C = I
& D = J
& E = K
& F = L ) ) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
fof(rc3_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) ) ).
fof(redefinition_k10_cat_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> k10_cat_1(A,B) = k5_cat_1(A,B) ) ).
fof(redefinition_k13_cat_2,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_subset_1(C,u2_cat_1(A))
& m1_subset_1(D,u2_cat_1(B)) )
=> k13_cat_2(A,B,C,D) = k4_tarski(C,D) ) ).
fof(redefinition_k1_domain_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,A)
& m1_subset_1(D,B) )
=> k1_domain_1(A,B,C,D) = k4_tarski(C,D) ) ).
fof(redefinition_k4_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> k4_relset_1(A,B,C) = k1_relat_1(C) ) ).
fof(redefinition_k7_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> k7_cat_2(A,B) = k5_cat_2(A,B) ) ).
fof(redefinition_k7_nattra_1,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B) )
=> k7_nattra_1(A,B,C) = k4_nattra_1(A,B,C) ) ).
fof(redefinition_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,A) )
=> k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
<=> m1_relset_1(C,A,B) ) ).
fof(redefinition_r4_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,A,B)
& m1_relset_1(E,A,B)
& v1_funct_1(F)
& v1_funct_2(F,C,D)
& m1_relset_1(F,C,D) )
=> ( r4_nattra_1(A,B,C,D,E,F)
<=> E = F ) ) ).
fof(reflexivity_r1_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> r1_nattra_1(A,B,C,C) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(reflexivity_r4_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,A,B)
& m1_relset_1(E,A,B)
& v1_funct_1(F)
& v1_funct_2(F,C,D)
& m1_relset_1(F,C,D) )
=> r4_nattra_1(A,B,C,D,E,E) ) ).
fof(symmetry_r4_nattra_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,A,B)
& m1_relset_1(E,A,B)
& v1_funct_1(F)
& v1_funct_2(F,C,D)
& m1_relset_1(F,C,D) )
=> ( r4_nattra_1(A,B,C,D,E,F)
=> r4_nattra_1(A,B,C,D,F,E) ) ) ).
fof(t15_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,B,A)
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> k3_nattra_1(B,A,C,D,D,k10_cat_1(B,D)) = k10_cat_1(A,k13_cat_1(B,A,C,D)) ) ) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t21_nattra_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,B,A)
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> k5_nattra_1(B,A,C,C,k4_nattra_1(B,A,C),D) = k10_cat_1(A,k13_cat_1(B,A,C,D)) ) ) ) ) ).
fof(t23_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_cat_1(D,A,B,C)
=> ( k6_cat_1(A,B,C) != k1_xboole_0
=> ( k2_cat_1(A,D) = B
& k3_cat_1(A,D) = C ) ) ) ) ) ) ).
fof(t28_mcart_1,axiom,
! [A,B,C,D,E,F] :
( k3_mcart_1(A,B,C) = k3_mcart_1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ).
fof(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t31_oppcat_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_oppcat_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(B))
=> ( k8_funct_2(u2_cat_1(A),u2_cat_1(B),C,k10_cat_1(A,D)) = k10_cat_1(B,E)
=> k8_funct_2(u1_cat_1(A),u1_cat_1(B),k12_cat_1(A,B,C),D) = E ) ) ) ) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t44_cat_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ( k2_cat_1(A,k5_cat_1(A,B)) = B
& k3_cat_1(A,k5_cat_1(A,B)) = B ) ) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) ).
fof(t53_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ( k22_ens_1(A,k10_cat_1(A,B),C) = k18_ens_1(A,B,C)
& k22_ens_1(A,C,k10_cat_1(A,B)) = k19_ens_1(A,B,C) ) ) ) ) ).
fof(t55_cat_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> r2_hidden(k5_cat_1(A,B),k6_cat_1(A,B,B)) ) ) ).
fof(t56_cat_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> k6_cat_1(A,B,B) != k1_xboole_0 ) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) ).
fof(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t6_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_oppcat_1(B,A,k12_nattra_1(A,k1_yoneda_1(A)))
=> ( ( v2_funct_1(k12_cat_1(A,k12_nattra_1(A,k1_yoneda_1(A)),B))
& v1_yoneda_1(B,A,k12_nattra_1(A,k1_yoneda_1(A))) )
=> v2_funct_1(B) ) ) ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) ).
fof(t7_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> v1_yoneda_1(k4_yoneda_1(A),A,k12_nattra_1(A,k1_yoneda_1(A))) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
fof(t9_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(A,C) = k1_funct_1(B,C) ) )
=> A = B ) ) ) ).
%------------------------------------------------------------------------------