SET007 Axioms: SET007+125.ax
%------------------------------------------------------------------------------
% File : SET007+125 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Cartesian Product of Functions
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : funct_6 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 114 ( 17 unt; 0 def)
% Number of atoms : 574 ( 155 equ)
% Maximal formula atoms : 18 ( 5 avg)
% Number of connectives : 493 ( 33 ~; 11 |; 243 &)
% ( 17 <=>; 189 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 15 ( 13 usr; 1 prp; 0-3 aty)
% Number of functors : 52 ( 52 usr; 6 con; 0-3 aty)
% Number of variables : 303 ( 292 !; 11 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_funct_6,axiom,
! [A,B] :
( r2_hidden(A,k4_card_3(k9_finseq_1(B)))
<=> ? [C] :
( r2_hidden(C,B)
& A = k9_finseq_1(C) ) ) ).
fof(t2_funct_6,axiom,
! [A,B,C] :
( r2_hidden(A,k4_card_3(k10_finseq_1(B,C)))
<=> ? [D,E] :
( r2_hidden(D,B)
& r2_hidden(E,C)
& A = k10_finseq_1(D,E) ) ) ).
fof(t3_funct_6,axiom,
! [A,B,C,D] :
( r2_hidden(A,k4_card_3(k11_finseq_1(B,C,D)))
<=> ? [E,F,G] :
( r2_hidden(E,B)
& r2_hidden(F,C)
& r2_hidden(G,D)
& A = k11_finseq_1(E,F,G) ) ) ).
fof(t4_funct_6,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_card_3(k9_finseq_1(A)) = k4_finseq_2(np__1,A) ) ).
fof(t6_funct_6,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_card_3(k10_finseq_1(A,A)) = k4_finseq_2(np__2,A) ) ).
fof(t8_funct_6,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_card_3(k11_finseq_1(A,A,A)) = k4_finseq_2(np__3,A) ) ).
fof(t9_funct_6,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> k4_card_3(k2_finseq_2(A,B)) = k4_finseq_2(A,B) ) ) ).
fof(t10_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> r1_tarski(k4_card_3(A),k1_funct_2(k1_relat_1(A),k3_card_3(A))) ) ).
fof(t11_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ~ ( r2_hidden(A,k1_relat_1(k2_funct_4(B)))
& ! [C,D] : A != k4_tarski(C,D) ) ) ).
fof(t12_funct_6,axiom,
! [A,B,C] : k2_funct_4(k2_funcop_1(k2_zfmisc_1(A,B),C)) = k2_funcop_1(k2_zfmisc_1(B,A),C) ).
fof(t13_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( k3_funct_5(A) = k5_funct_5(k2_funct_4(A))
& k4_funct_5(A) = k2_funct_4(k6_funct_5(A)) ) ) ).
fof(t14_funct_6,axiom,
! [A,B,C] :
( k2_zfmisc_1(A,B) != k1_xboole_0
=> ( k3_funct_5(k2_funcop_1(k2_zfmisc_1(A,B),C)) = k2_funcop_1(A,k2_funcop_1(B,C))
& k5_funct_5(k2_funcop_1(k2_zfmisc_1(A,B),C)) = k2_funcop_1(B,k2_funcop_1(A,C)) ) ) ).
fof(t15_funct_6,axiom,
! [A,B,C] :
( k4_funct_5(k2_funcop_1(A,k2_funcop_1(B,C))) = k2_funcop_1(k2_zfmisc_1(A,B),C)
& k6_funct_5(k2_funcop_1(A,k2_funcop_1(B,C))) = k2_funcop_1(k2_zfmisc_1(B,A),C) ) ).
fof(t16_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(A,k1_relat_1(B))
& C = k1_funct_1(B,A) )
=> ( r1_tarski(k2_relat_1(C),k2_relat_1(k4_funct_5(B)))
& r1_tarski(k2_relat_1(C),k2_relat_1(k6_funct_5(B))) ) ) ) ) ).
fof(t17_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( k1_relat_1(k4_funct_5(k2_funcop_1(A,B))) = k2_zfmisc_1(A,k1_relat_1(B))
& r1_tarski(k2_relat_1(k4_funct_5(k2_funcop_1(A,B))),k2_relat_1(B))
& k1_relat_1(k6_funct_5(k2_funcop_1(A,B))) = k2_zfmisc_1(k1_relat_1(B),A)
& r1_tarski(k2_relat_1(k6_funct_5(k2_funcop_1(A,B))),k2_relat_1(B)) ) ) ).
fof(t18_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( A != k1_xboole_0
=> ( k2_relat_1(k4_funct_5(k2_funcop_1(A,B))) = k2_relat_1(B)
& k2_relat_1(k6_funct_5(k2_funcop_1(A,B))) = k2_relat_1(B) ) ) ) ).
fof(t19_funct_6,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(D,k1_funct_2(k2_zfmisc_1(A,B),C))
=> ( k2_zfmisc_1(A,B) = k1_xboole_0
| ( r2_hidden(k3_funct_5(D),k1_funct_2(A,k1_funct_2(B,C)))
& r2_hidden(k5_funct_5(D),k1_funct_2(B,k1_funct_2(A,C))) ) ) ) ) ).
fof(t20_funct_6,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(D,k1_funct_2(A,k1_funct_2(B,C)))
=> ( r2_hidden(k4_funct_5(D),k1_funct_2(k2_zfmisc_1(A,B),C))
& r2_hidden(k6_funct_5(D),k1_funct_2(k2_zfmisc_1(B,A),C)) ) ) ) ).
fof(t21_funct_6,axiom,
! [A,B,C,D,E,F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ( r1_tarski(k1_relat_1(F),k2_zfmisc_1(D,E))
=> ( ( ~ r2_hidden(k3_funct_5(F),k1_funct_2(A,k1_funct_2(B,C)))
& ~ r2_hidden(k5_funct_5(F),k1_funct_2(B,k1_funct_2(A,C))) )
| r2_hidden(F,k1_funct_2(k2_zfmisc_1(A,B),C)) ) ) ) ).
fof(t22_funct_6,axiom,
! [A,B,C,D,E,F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ( ( r1_tarski(k2_relat_1(F),k4_partfun1(D,E))
& k1_relat_1(F) = A )
=> ( ( ~ r2_hidden(k4_funct_5(F),k1_funct_2(k2_zfmisc_1(A,B),C))
& ~ r2_hidden(k6_funct_5(F),k1_funct_2(k2_zfmisc_1(B,A),C)) )
| r2_hidden(F,k1_funct_2(A,k1_funct_2(B,C))) ) ) ) ).
fof(t23_funct_6,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(D,k4_partfun1(k2_zfmisc_1(A,B),C))
=> ( r2_hidden(k3_funct_5(D),k4_partfun1(A,k4_partfun1(B,C)))
& r2_hidden(k5_funct_5(D),k4_partfun1(B,k4_partfun1(A,C))) ) ) ) ).
fof(t24_funct_6,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(D,k4_partfun1(A,k4_partfun1(B,C)))
=> ( r2_hidden(k4_funct_5(D),k4_partfun1(k2_zfmisc_1(A,B),C))
& r2_hidden(k6_funct_5(D),k4_partfun1(k2_zfmisc_1(B,A),C)) ) ) ) ).
fof(t25_funct_6,axiom,
! [A,B,C,D,E,F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ( r1_tarski(k1_relat_1(F),k2_zfmisc_1(D,E))
=> ( ( ~ r2_hidden(k3_funct_5(F),k4_partfun1(A,k4_partfun1(B,C)))
& ~ r2_hidden(k5_funct_5(F),k4_partfun1(B,k4_partfun1(A,C))) )
| r2_hidden(F,k4_partfun1(k2_zfmisc_1(A,B),C)) ) ) ) ).
fof(t26_funct_6,axiom,
! [A,B,C,D,E,F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ( ( r1_tarski(k2_relat_1(F),k4_partfun1(D,E))
& r1_tarski(k1_relat_1(F),A) )
=> ( ( ~ r2_hidden(k4_funct_5(F),k4_partfun1(k2_zfmisc_1(A,B),C))
& ~ r2_hidden(k6_funct_5(F),k4_partfun1(k2_zfmisc_1(B,A),C)) )
| r2_hidden(F,k4_partfun1(A,k4_partfun1(B,C))) ) ) ) ).
fof(d1_funct_6,axiom,
! [A,B] :
( B = k1_funct_6(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( r2_hidden(C,A)
& v1_relat_1(C)
& v1_funct_1(C) ) ) ) ).
fof(t27_funct_6,axiom,
! [A] : r1_tarski(k1_funct_6(A),A) ).
fof(t28_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k10_relat_1(B,k1_funct_6(k2_relat_1(B))))
<=> ( r2_hidden(A,k1_relat_1(B))
& v1_relat_1(k1_funct_1(B,A))
& v1_funct_1(k1_funct_1(B,A)) ) ) ) ).
fof(t29_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k1_funct_6(k1_xboole_0) = k1_xboole_0
& k1_funct_6(k1_tarski(A)) = k1_tarski(A)
& k1_funct_6(k2_tarski(A,B)) = k2_tarski(A,B)
& k1_funct_6(k1_enumset1(A,B,C)) = k1_enumset1(A,B,C) ) ) ) ) ).
fof(t30_funct_6,axiom,
! [A,B] :
( r1_tarski(A,k1_funct_6(B))
=> k1_funct_6(A) = A ) ).
fof(d2_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k2_funct_6(A)
<=> ( k1_relat_1(B) = k10_relat_1(A,k1_funct_6(k2_relat_1(A)))
& ! [C] :
( r2_hidden(C,k10_relat_1(A,k1_funct_6(k2_relat_1(A))))
=> k1_funct_1(B,C) = k1_funct_5(k1_funct_1(A,C)) ) ) ) ) ) ).
fof(d3_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k3_funct_6(A)
<=> ( k1_relat_1(B) = k10_relat_1(A,k1_funct_6(k2_relat_1(A)))
& ! [C] :
( r2_hidden(C,k10_relat_1(A,k1_funct_6(k2_relat_1(A))))
=> k1_funct_1(B,C) = k2_funct_5(k1_funct_1(A,C)) ) ) ) ) ) ).
fof(d4_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k4_funct_6(A) = k1_setfam_1(k2_relat_1(A)) ) ).
fof(t31_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(A,k1_relat_1(B))
& C = k1_funct_1(B,A) )
=> ( r2_hidden(A,k1_relat_1(k2_funct_6(B)))
& k1_funct_1(k2_funct_6(B),A) = k1_relat_1(C)
& r2_hidden(A,k1_relat_1(k3_funct_6(B)))
& k1_funct_1(k3_funct_6(B),A) = k2_relat_1(C) ) ) ) ) ).
fof(t32_funct_6,axiom,
( k2_funct_6(k1_xboole_0) = k1_xboole_0
& k3_funct_6(k1_xboole_0) = k1_xboole_0 ) ).
fof(t33_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( k2_funct_6(k9_finseq_1(A)) = k9_finseq_1(k1_relat_1(A))
& k3_funct_6(k9_finseq_1(A)) = k9_finseq_1(k2_relat_1(A)) ) ) ).
fof(t34_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( k2_funct_6(k10_finseq_1(A,B)) = k10_finseq_1(k1_relat_1(A),k1_relat_1(B))
& k3_funct_6(k10_finseq_1(A,B)) = k10_finseq_1(k2_relat_1(A),k2_relat_1(B)) ) ) ) ).
fof(t35_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k2_funct_6(k11_finseq_1(A,B,C)) = k11_finseq_1(k1_relat_1(A),k1_relat_1(B),k1_relat_1(C))
& k3_funct_6(k11_finseq_1(A,B,C)) = k11_finseq_1(k2_relat_1(A),k2_relat_1(B),k2_relat_1(C)) ) ) ) ) ).
fof(t36_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( k2_funct_6(k2_funcop_1(A,B)) = k2_funcop_1(A,k1_relat_1(B))
& k3_funct_6(k2_funcop_1(A,B)) = k2_funcop_1(A,k2_relat_1(B)) ) ) ).
fof(t37_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B != k1_xboole_0
=> ( r2_hidden(A,k4_funct_6(B))
<=> ! [C] :
( r2_hidden(C,k1_relat_1(B))
=> r2_hidden(A,k1_funct_1(B,C)) ) ) ) ) ).
fof(t38_funct_6,axiom,
k4_funct_6(k1_xboole_0) = k1_xboole_0 ).
fof(t39_funct_6,axiom,
! [A] :
( k3_card_3(k9_finseq_1(A)) = A
& k4_funct_6(k9_finseq_1(A)) = A ) ).
fof(t40_funct_6,axiom,
! [A,B] :
( k3_card_3(k10_finseq_1(A,B)) = k2_xboole_0(A,B)
& k4_funct_6(k10_finseq_1(A,B)) = k3_xboole_0(A,B) ) ).
fof(t41_funct_6,axiom,
! [A,B,C] :
( k3_card_3(k11_finseq_1(A,B,C)) = k2_xboole_0(k2_xboole_0(A,B),C)
& k4_funct_6(k11_finseq_1(A,B,C)) = k3_xboole_0(k3_xboole_0(A,B),C) ) ).
fof(t42_funct_6,axiom,
! [A] :
( k3_card_3(k2_funcop_1(k1_xboole_0,A)) = k1_xboole_0
& k4_funct_6(k2_funcop_1(k1_xboole_0,A)) = k1_xboole_0 ) ).
fof(t43_funct_6,axiom,
! [A,B] :
( A != k1_xboole_0
=> ( k3_card_3(k2_funcop_1(A,B)) = B
& k4_funct_6(k2_funcop_1(A,B)) = B ) ) ).
fof(d5_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] : k5_funct_6(A,B,C) = k1_funct_1(k4_funct_5(A),k4_tarski(B,C)) ) ).
fof(t44_funct_6,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r2_hidden(A,k1_relat_1(C))
& D = k1_funct_1(C,A)
& r2_hidden(B,k1_relat_1(D)) )
=> k5_funct_6(C,A,B) = k1_funct_1(D,B) ) ) ) ).
fof(t45_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(A,k1_relat_1(B))
=> ( k5_funct_6(k9_finseq_1(B),np__1,A) = k1_funct_1(B,A)
& k5_funct_6(k10_finseq_1(B,C),np__1,A) = k1_funct_1(B,A)
& k5_funct_6(k11_finseq_1(B,C,D),np__1,A) = k1_funct_1(B,A) ) ) ) ) ) ).
fof(t46_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(A,k1_relat_1(B))
=> ( k5_funct_6(k10_finseq_1(C,B),np__2,A) = k1_funct_1(B,A)
& k5_funct_6(k11_finseq_1(C,B,D),np__2,A) = k1_funct_1(B,A) ) ) ) ) ) ).
fof(t47_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(A,k1_relat_1(B))
=> k5_funct_6(k11_finseq_1(C,D,B),np__3,A) = k1_funct_1(B,A) ) ) ) ) ).
fof(t48_funct_6,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r2_hidden(A,B)
& r2_hidden(C,k1_relat_1(D)) )
=> k5_funct_6(k2_funcop_1(B,D),A,C) = k1_funct_1(D,C) ) ) ).
fof(d6_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k6_funct_6(A) = k3_funct_5(k7_relat_1(k6_funct_5(A),k2_zfmisc_1(k4_funct_6(k2_funct_6(A)),k1_relat_1(A)))) ) ).
fof(t49_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( k1_relat_1(k6_funct_6(A)) = k4_funct_6(k2_funct_6(A))
& r1_tarski(k2_relat_1(k6_funct_6(A)),k4_card_3(k3_funct_6(A))) ) ) ).
fof(t50_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k1_relat_1(k6_funct_6(B)))
=> ( v1_relat_1(k1_funct_1(k6_funct_6(B),A))
& v1_funct_1(k1_funct_1(k6_funct_6(B),A)) ) ) ) ).
fof(t51_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(A,k1_relat_1(k6_funct_6(B)))
& C = k1_funct_1(k6_funct_6(B),A) )
=> ( k1_relat_1(C) = k10_relat_1(B,k1_funct_6(k2_relat_1(B)))
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> ( r2_hidden(k4_tarski(D,A),k1_relat_1(k4_funct_5(B)))
& k1_funct_1(C,D) = k1_funct_1(k4_funct_5(B),k4_tarski(D,A)) ) ) ) ) ) ) ).
fof(t52_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k1_relat_1(k6_funct_6(B)))
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(C,k2_relat_1(B))
=> r2_hidden(A,k1_relat_1(C)) ) ) ) ) ).
fof(t53_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(B,k2_relat_1(C))
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(D,k2_relat_1(C))
=> r2_hidden(A,k1_relat_1(D)) ) ) )
=> r2_hidden(A,k1_relat_1(k6_funct_6(C))) ) ) ) ).
fof(t54_funct_6,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r2_hidden(A,k1_relat_1(C))
& D = k1_funct_1(C,A)
& r2_hidden(B,k1_relat_1(k6_funct_6(C)))
& E = k1_funct_1(k6_funct_6(C),B) )
=> k1_funct_1(D,B) = k1_funct_1(E,A) ) ) ) ) ).
fof(t55_funct_6,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(A,k1_relat_1(C))
& v1_relat_1(k1_funct_1(C,A))
& v1_funct_1(k1_funct_1(C,A))
& r2_hidden(B,k1_relat_1(k6_funct_6(C))) )
=> k5_funct_6(C,A,B) = k5_funct_6(k6_funct_6(C),B,A) ) ) ).
fof(d7_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k7_funct_6(A)
<=> ( k1_relat_1(B) = k4_card_3(k2_funct_6(A))
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ~ ( r2_hidden(C,k4_card_3(k2_funct_6(A)))
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ~ ( k1_funct_1(B,C) = D
& k1_relat_1(D) = k10_relat_1(A,k1_funct_6(k2_relat_1(A)))
& ! [E] :
( r2_hidden(E,k1_relat_1(D))
=> k1_funct_1(D,E) = k1_funct_1(k4_funct_5(A),k4_tarski(E,k1_funct_1(C,E))) ) ) ) ) ) ) ) ) ) ).
fof(t56_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(B,k4_card_3(k2_funct_6(C)))
& r2_hidden(A,k1_relat_1(B)) )
=> k5_funct_6(k7_funct_6(C),B,A) = k5_funct_6(C,A,k1_funct_1(B,A)) ) ) ) ).
fof(t57_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r2_hidden(A,k1_relat_1(B))
& C = k1_funct_1(B,A)
& r2_hidden(D,k4_card_3(k2_funct_6(B)))
& E = k1_funct_1(k7_funct_6(B),D) )
=> ( r2_hidden(k1_funct_1(D,A),k1_relat_1(C))
& k1_funct_1(E,A) = k1_funct_1(C,k1_funct_1(D,A))
& r2_hidden(E,k4_card_3(k3_funct_6(B))) ) ) ) ) ) ) ).
fof(t58_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k2_relat_1(k7_funct_6(A)) = k4_card_3(k3_funct_6(A)) ) ).
fof(t59_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ~ r2_hidden(k1_xboole_0,k2_relat_1(A))
=> ( v2_funct_1(k7_funct_6(A))
<=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(B,k2_relat_1(A))
=> v2_funct_1(B) ) ) ) ) ) ).
fof(t60_funct_6,axiom,
( k6_funct_6(k1_xboole_0) = k1_xboole_0
& k7_funct_6(k1_xboole_0) = k2_funcop_1(k1_tarski(k1_xboole_0),k1_xboole_0) ) ).
fof(t61_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( k1_relat_1(k6_funct_6(k9_finseq_1(A))) = k1_relat_1(A)
& ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> k1_funct_1(k6_funct_6(k9_finseq_1(A)),B) = k9_finseq_1(k1_funct_1(A,B)) ) ) ) ).
fof(t62_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( k1_relat_1(k6_funct_6(k10_finseq_1(A,B))) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
& ! [C] :
( r2_hidden(C,k3_xboole_0(k1_relat_1(A),k1_relat_1(B)))
=> k1_funct_1(k6_funct_6(k10_finseq_1(A,B)),C) = k10_finseq_1(k1_funct_1(A,C),k1_funct_1(B,C)) ) ) ) ) ).
fof(t63_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( A != k1_xboole_0
=> ( k1_relat_1(k6_funct_6(k2_funcop_1(A,B))) = k1_relat_1(B)
& ! [C] :
( r2_hidden(C,k1_relat_1(B))
=> k1_funct_1(k6_funct_6(k2_funcop_1(A,B)),C) = k2_funcop_1(A,k1_funct_1(B,C)) ) ) ) ) ).
fof(t64_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( k1_relat_1(k7_funct_6(k9_finseq_1(A))) = k4_card_3(k9_finseq_1(k1_relat_1(A)))
& k2_relat_1(k7_funct_6(k9_finseq_1(A))) = k4_card_3(k9_finseq_1(k2_relat_1(A)))
& ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> k1_funct_1(k7_funct_6(k9_finseq_1(A)),k9_finseq_1(B)) = k9_finseq_1(k1_funct_1(A,B)) ) ) ) ).
fof(t65_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( k1_relat_1(k7_funct_6(k10_finseq_1(A,B))) = k4_card_3(k10_finseq_1(k1_relat_1(A),k1_relat_1(B)))
& k2_relat_1(k7_funct_6(k10_finseq_1(A,B))) = k4_card_3(k10_finseq_1(k2_relat_1(A),k2_relat_1(B)))
& ! [C,D] :
( ( r2_hidden(C,k1_relat_1(A))
& r2_hidden(D,k1_relat_1(B)) )
=> k1_funct_1(k7_funct_6(k10_finseq_1(A,B)),k10_finseq_1(C,D)) = k10_finseq_1(k1_funct_1(A,C),k1_funct_1(B,D)) ) ) ) ) ).
fof(t66_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( k1_relat_1(k7_funct_6(k2_funcop_1(A,B))) = k1_funct_2(A,k1_relat_1(B))
& k2_relat_1(k7_funct_6(k2_funcop_1(A,B))) = k1_funct_2(A,k2_relat_1(B))
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(C,k1_funct_2(A,k1_relat_1(B)))
=> k1_funct_1(k7_funct_6(k2_funcop_1(A,B)),C) = k5_relat_1(C,B) ) ) ) ) ).
fof(t67_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(A,k1_relat_1(B))
& r2_hidden(A,k1_relat_1(C)) )
=> ! [D,E] :
( k1_funct_1(k13_funct_3(B,C),A) = k4_tarski(D,E)
<=> k1_funct_1(k6_funct_6(k10_finseq_1(B,C)),A) = k10_finseq_1(D,E) ) ) ) ) ).
fof(t68_funct_6,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r2_hidden(A,k1_relat_1(C))
& r2_hidden(B,k1_relat_1(D)) )
=> ! [E,F] :
( k1_funct_1(k15_funct_3(C,D),k4_tarski(A,B)) = k4_tarski(E,F)
<=> k1_funct_1(k7_funct_6(k10_finseq_1(C,D)),k10_finseq_1(A,B)) = k10_finseq_1(E,F) ) ) ) ) ).
fof(t69_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( k1_relat_1(B) = A
& k1_relat_1(C) = A
& ! [D] :
( r2_hidden(D,A)
=> r2_wellord2(k1_funct_1(B,D),k1_funct_1(C,D)) ) )
=> r2_wellord2(k4_card_3(B),k4_card_3(C)) ) ) ) ).
fof(t70_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& k1_relat_1(C) = k2_relat_1(B)
& v2_funct_1(B)
& ! [D] :
( r2_hidden(D,k1_relat_1(B))
=> r2_wellord2(k1_funct_1(A,D),k1_funct_1(C,k1_funct_1(B,D))) ) )
=> r2_wellord2(k4_card_3(A),k4_card_3(C)) ) ) ) ) ).
fof(t71_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& v3_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> ( k1_relat_1(B) = A
=> r2_wellord2(k4_card_3(B),k4_card_3(k5_relat_1(C,B))) ) ) ) ).
fof(d8_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k8_funct_6(A,B)
<=> ( k1_relat_1(C) = k1_relat_1(A)
& ! [D] :
( r2_hidden(D,k1_relat_1(A))
=> k1_funct_1(C,D) = k1_funct_2(k1_funct_1(A,D),B) ) ) ) ) ) ).
fof(t72_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ~ r2_hidden(k1_xboole_0,k2_relat_1(A))
=> k8_funct_6(A,k1_xboole_0) = k2_funcop_1(k1_relat_1(A),k1_xboole_0) ) ) ).
fof(t73_funct_6,axiom,
! [A] : k8_funct_6(k1_xboole_0,A) = k1_xboole_0 ).
fof(t74_funct_6,axiom,
! [A,B] : k8_funct_6(k9_finseq_1(A),B) = k9_finseq_1(k1_funct_2(A,B)) ).
fof(t75_funct_6,axiom,
! [A,B,C] : k8_funct_6(k10_finseq_1(A,B),C) = k10_finseq_1(k1_funct_2(A,C),k1_funct_2(B,C)) ).
fof(t76_funct_6,axiom,
! [A,B,C] : k8_funct_6(k2_funcop_1(A,B),C) = k2_funcop_1(A,k1_funct_2(B,C)) ).
fof(t77_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r2_wellord2(k1_funct_2(k3_card_3(k2_card_3(B)),A),k4_card_3(k8_funct_6(B,A))) ) ).
fof(d9_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k9_funct_6(A,B)
<=> ( k1_relat_1(C) = k1_relat_1(B)
& ! [D] :
( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(C,D) = k1_funct_2(A,k1_funct_1(B,D)) ) ) ) ) ) ).
fof(t78_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k9_funct_6(k1_xboole_0,A) = k2_funcop_1(k1_relat_1(A),k1_tarski(k1_xboole_0)) ) ).
fof(t79_funct_6,axiom,
! [A] : k9_funct_6(A,k1_xboole_0) = k1_xboole_0 ).
fof(t80_funct_6,axiom,
! [A,B] : k9_funct_6(A,k9_finseq_1(B)) = k9_finseq_1(k1_funct_2(A,B)) ).
fof(t81_funct_6,axiom,
! [A,B,C] : k9_funct_6(A,k10_finseq_1(B,C)) = k10_finseq_1(k1_funct_2(A,B),k1_funct_2(A,C)) ).
fof(t82_funct_6,axiom,
! [A,B,C] : k9_funct_6(A,k2_funcop_1(B,C)) = k2_funcop_1(B,k1_funct_2(A,C)) ).
fof(t83_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r2_wellord2(k4_card_3(k9_funct_6(A,B)),k1_funct_2(A,k4_card_3(B))) ) ).
fof(d10_funct_6,axiom,
$true ).
fof(d11_funct_6,axiom,
$true ).
fof(d12_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k10_funct_6(A) = k5_funct_5(k4_funct_5(A)) ) ).
fof(t84_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_hidden(B,k1_relat_1(k10_funct_6(A)))
=> ( v1_relat_1(k1_funct_1(k10_funct_6(A),B))
& v1_funct_1(k1_funct_1(k10_funct_6(A),B)) ) ) ) ).
fof(t85_funct_6,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(D,k1_funct_2(A,k1_funct_2(B,C)))
=> ( A = k1_xboole_0
| B = k1_xboole_0
| r2_hidden(k10_funct_6(D),k1_funct_2(B,k1_funct_2(A,C))) ) ) ) ).
fof(t86_funct_6,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(D,k1_funct_2(A,k1_funct_2(B,C)))
=> ( A = k1_xboole_0
| B = k1_xboole_0
| ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ! [G,H] :
( ( r2_hidden(G,A)
& r2_hidden(H,B)
& k1_funct_1(D,G) = E
& k1_funct_1(k10_funct_6(D),H) = F )
=> ( k1_funct_1(F,G) = k1_funct_1(E,H)
& k1_relat_1(F) = A
& k1_relat_1(E) = B
& r1_tarski(k2_relat_1(F),C)
& r1_tarski(k2_relat_1(E),C) ) ) ) ) ) ) ) ).
fof(t87_funct_6,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(D,k1_funct_2(A,k1_funct_2(B,C)))
=> ( A = k1_xboole_0
| B = k1_xboole_0
| k10_funct_6(k10_funct_6(D)) = D ) ) ) ).
fof(t88_funct_6,axiom,
k10_funct_6(k1_xboole_0) = k1_xboole_0 ).
fof(t89_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> k1_relat_1(k2_funct_6(A)) = k1_relat_1(A) ) ).
fof(t90_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> k1_relat_1(k3_funct_6(A)) = k1_relat_1(A) ) ).
fof(dt_k1_funct_6,axiom,
$true ).
fof(dt_k2_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k2_funct_6(A))
& v1_funct_1(k2_funct_6(A)) ) ) ).
fof(dt_k3_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k3_funct_6(A))
& v1_funct_1(k3_funct_6(A)) ) ) ).
fof(dt_k4_funct_6,axiom,
$true ).
fof(dt_k5_funct_6,axiom,
$true ).
fof(dt_k6_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k6_funct_6(A))
& v1_funct_1(k6_funct_6(A)) ) ) ).
fof(dt_k7_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k7_funct_6(A))
& v1_funct_1(k7_funct_6(A)) ) ) ).
fof(dt_k8_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k8_funct_6(A,B))
& v1_funct_1(k8_funct_6(A,B)) ) ) ).
fof(dt_k9_funct_6,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k9_funct_6(A,B))
& v1_funct_1(k9_funct_6(A,B)) ) ) ).
fof(dt_k10_funct_6,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k10_funct_6(A))
& v1_funct_1(k10_funct_6(A))
& v1_funcop_1(k10_funct_6(A)) ) ) ).
fof(t5_funct_6,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> k4_card_3(k10_finseq_1(A,B)) = a_2_0_funct_6(A,B) ) ) ).
fof(t7_funct_6,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> k4_card_3(k11_finseq_1(A,B,C)) = a_3_0_funct_6(A,B,C) ) ) ) ).
fof(fraenkel_a_2_0_funct_6,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C) )
=> ( r2_hidden(A,a_2_0_funct_6(B,C))
<=> ? [D,E] :
( m1_subset_1(D,B)
& m1_subset_1(E,C)
& A = k10_finseq_1(D,E) ) ) ) ).
fof(fraenkel_a_3_0_funct_6,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D) )
=> ( r2_hidden(A,a_3_0_funct_6(B,C,D))
<=> ? [E,F,G] :
( m1_subset_1(E,B)
& m1_subset_1(F,C)
& m1_subset_1(G,D)
& A = k11_finseq_1(E,F,G) ) ) ) ).
%------------------------------------------------------------------------------