SET007 Axioms: SET007+102.ax
%------------------------------------------------------------------------------
% File : SET007+102 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Partial Functions from a Domain to a Domain
% 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 : partfun2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 95 ( 22 unt; 0 def)
% Number of atoms : 666 ( 91 equ)
% Maximal formula atoms : 19 ( 7 avg)
% Number of connectives : 709 ( 138 ~; 1 |; 191 &)
% ( 31 <=>; 348 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 19 ( 17 usr; 1 prp; 0-4 aty)
% Number of functors : 37 ( 37 usr; 8 con; 0-6 aty)
% Number of variables : 351 ( 340 !; 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(fc1_partfun2,axiom,
( v1_xboole_0(k1_xboole_0)
& v1_relat_1(k1_xboole_0)
& v3_relat_1(k1_xboole_0)
& v1_funct_1(k1_xboole_0)
& v2_funct_1(k1_xboole_0) ) ).
fof(rc1_partfun2,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C)
& v2_funct_1(C) ) ).
fof(t1_partfun2,axiom,
$true ).
fof(t2_partfun2,axiom,
$true ).
fof(t3_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,B,A) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,A) )
=> ( ( k4_relset_1(B,A,C) = k4_relset_1(B,A,D)
& ! [E] :
( m1_subset_1(E,B)
=> ( r2_hidden(E,k4_relset_1(B,A,C))
=> k4_finseq_4(B,A,C,E) = k4_finseq_4(B,A,D,E) ) ) )
=> C = D ) ) ) ) ) ).
fof(t4_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,C,B) )
=> ( r2_hidden(A,k2_relat_1(D))
<=> ? [E] :
( m1_subset_1(E,C)
& r2_hidden(E,k4_relset_1(C,B,D))
& A = k4_finseq_4(C,B,D,E) ) ) ) ) ) ).
fof(t5_partfun2,axiom,
$true ).
fof(t6_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,C,A) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,C,B) )
=> ( F = k1_partfun1(C,A,A,B,D,E)
<=> ( ! [G] :
( m1_subset_1(G,C)
=> ( r2_hidden(G,k4_relset_1(C,B,F))
<=> ( r2_hidden(G,k4_relset_1(C,A,D))
& r2_hidden(k4_finseq_4(C,A,D,G),k4_relset_1(A,B,E)) ) ) )
& ! [G] :
( m1_subset_1(G,C)
=> ( r2_hidden(G,k4_relset_1(C,B,F))
=> k4_finseq_4(C,B,F,G) = k4_finseq_4(A,B,E,k4_finseq_4(C,A,D,G)) ) ) ) ) ) ) ) ) ) ) ).
fof(t7_partfun2,axiom,
$true ).
fof(t8_partfun2,axiom,
$true ).
fof(t9_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,B,C) )
=> ( ( r2_hidden(D,k4_relset_1(A,B,E))
& r2_hidden(k4_finseq_4(A,B,E,D),k4_relset_1(B,C,F)) )
=> k4_finseq_4(A,C,k1_partfun1(A,B,B,C,E,F),D) = k4_finseq_4(B,C,F,k4_finseq_4(A,B,E,D)) ) ) ) ) ) ) ) ).
fof(t10_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,B,C) )
=> ( ( r1_tarski(k2_relat_1(E),k4_relset_1(B,C,F))
& r2_hidden(D,k4_relset_1(A,B,E)) )
=> k4_finseq_4(A,C,k1_partfun1(A,B,B,C,E,F),D) = k4_finseq_4(B,C,F,k4_finseq_4(A,B,E,D)) ) ) ) ) ) ) ) ).
fof(t11_partfun2,axiom,
$true ).
fof(t12_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,A) )
=> ( C = k1_partfun2(A,B)
<=> ( k4_relset_1(A,A,C) = B
& ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,B)
=> k4_finseq_4(A,A,C,D) = D ) ) ) ) ) ) ) ).
fof(t13_partfun2,axiom,
$true ).
fof(t14_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,A) )
=> ( r2_hidden(C,k5_subset_1(A,k4_relset_1(A,A,D),B))
=> k4_finseq_4(A,A,D,C) = k4_finseq_4(A,A,k1_partfun1(A,A,A,A,k1_partfun2(A,B),D),C) ) ) ) ) ) ).
fof(t15_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,A) )
=> ( r2_hidden(C,k4_relset_1(A,A,k1_partfun1(A,A,A,A,D,k1_partfun2(A,B))))
<=> ( r2_hidden(C,k4_relset_1(A,A,D))
& r2_hidden(k4_finseq_4(A,A,D,C),B) ) ) ) ) ) ) ).
fof(t16_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,B,A) )
=> ( ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( m1_subset_1(E,B)
=> ( ( r2_hidden(D,k4_relset_1(B,A,C))
& r2_hidden(E,k4_relset_1(B,A,C))
& k4_finseq_4(B,A,C,D) = k4_finseq_4(B,A,C,E) )
=> D = E ) ) )
=> v2_funct_1(C) ) ) ) ) ).
fof(t17_partfun2,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,D) )
=> ( ( v2_funct_1(E)
& r2_hidden(A,k4_relset_1(C,D,E))
& r2_hidden(B,k4_relset_1(C,D,E))
& k4_finseq_4(C,D,E,A) = k4_finseq_4(C,D,E,B) )
=> A = B ) ) ) ) ).
fof(t18_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v2_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,A) )
=> ( D = k2_partfun2(A,B,C)
<=> ( k4_relset_1(B,A,D) = k2_relat_1(C)
& ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( m1_subset_1(F,A)
=> ( ( ( r2_hidden(E,k2_relat_1(C))
& F = k4_finseq_4(B,A,D,E) )
=> ( r2_hidden(F,k4_relset_1(A,B,C))
& E = k4_finseq_4(A,B,C,F) ) )
& ( ( r2_hidden(F,k4_relset_1(A,B,C))
& E = k4_finseq_4(A,B,C,F) )
=> ( r2_hidden(E,k2_relat_1(C))
& F = k4_finseq_4(B,A,D,E) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t19_partfun2,axiom,
$true ).
fof(t20_partfun2,axiom,
$true ).
fof(t21_partfun2,axiom,
$true ).
fof(t22_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v2_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( r2_hidden(C,k4_relset_1(A,B,D))
=> ( C = k4_finseq_4(B,A,k2_partfun2(A,B,D),k4_finseq_4(A,B,D,C))
& C = k4_finseq_4(A,A,k1_partfun1(A,B,B,A,D,k2_partfun2(A,B,D)),C) ) ) ) ) ) ) ).
fof(t23_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( ( v1_funct_1(D)
& v2_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( r2_hidden(C,k2_relat_1(D))
=> ( C = k4_finseq_4(A,B,D,k4_finseq_4(B,A,k2_partfun2(A,B,D),C))
& C = k4_finseq_4(B,B,k1_partfun1(B,A,A,B,k2_partfun2(A,B,D),D),C) ) ) ) ) ) ) ).
fof(t24_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,A) )
=> ( ( v2_funct_1(C)
& k4_relset_1(A,B,C) = k2_relat_1(D)
& k2_relat_1(C) = k4_relset_1(B,A,D)
& ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,B)
=> ( ( r2_hidden(E,k4_relset_1(A,B,C))
& r2_hidden(F,k4_relset_1(B,A,D)) )
=> ( k4_finseq_4(A,B,C,E) = F
<=> k4_finseq_4(B,A,D,F) = E ) ) ) ) )
=> D = k2_funct_1(C) ) ) ) ) ) ).
fof(t25_partfun2,axiom,
$true ).
fof(t26_partfun2,axiom,
$true ).
fof(t27_partfun2,axiom,
$true ).
fof(t28_partfun2,axiom,
$true ).
fof(t29_partfun2,axiom,
$true ).
fof(t30_partfun2,axiom,
$true ).
fof(t31_partfun2,axiom,
$true ).
fof(t32_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,C,B) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,B) )
=> ( D = k2_partfun1(C,B,E,A)
<=> ( k4_relset_1(C,B,D) = k3_xboole_0(k4_relset_1(C,B,E),A)
& ! [F] :
( m1_subset_1(F,C)
=> ( r2_hidden(F,k4_relset_1(C,B,D))
=> k4_finseq_4(C,B,D,F) = k4_finseq_4(C,B,E,F) ) ) ) ) ) ) ) ) ).
fof(t33_partfun2,axiom,
$true ).
fof(t34_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,B,C) )
=> ( r2_hidden(D,k3_xboole_0(k4_relset_1(B,C,E),A))
=> k4_finseq_4(B,C,k2_partfun1(B,C,E,A),D) = k4_finseq_4(B,C,E,D) ) ) ) ) ) ).
fof(t35_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,B,C) )
=> ( ( r2_hidden(D,k4_relset_1(B,C,E))
& r2_hidden(D,A) )
=> k4_finseq_4(B,C,k2_partfun1(B,C,E,A),D) = k4_finseq_4(B,C,E,D) ) ) ) ) ) ).
fof(t36_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,B,C) )
=> ( ( r2_hidden(D,k4_relset_1(B,C,E))
& r2_hidden(D,A) )
=> r2_hidden(k4_finseq_4(B,C,E,D),k2_relat_1(k2_partfun1(B,C,E,A))) ) ) ) ) ) ).
fof(t37_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,C,B) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,B) )
=> ( D = k3_partfun2(C,B,A,E)
<=> ( ! [F] :
( m1_subset_1(F,C)
=> ( r2_hidden(F,k4_relset_1(C,B,D))
<=> ( r2_hidden(F,k4_relset_1(C,B,E))
& r2_hidden(k4_finseq_4(C,B,E,F),A) ) ) )
& ! [F] :
( m1_subset_1(F,C)
=> ( r2_hidden(F,k4_relset_1(C,B,D))
=> k4_finseq_4(C,B,D,F) = k4_finseq_4(C,B,E,F) ) ) ) ) ) ) ) ) ).
fof(t38_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,B,C) )
=> ( r2_hidden(D,k4_relset_1(B,C,k3_partfun2(B,C,A,E)))
<=> ( r2_hidden(D,k4_relset_1(B,C,E))
& r2_hidden(k4_finseq_4(B,C,E,D),A) ) ) ) ) ) ) ).
fof(t39_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,B,C) )
=> ( r2_hidden(D,k4_relset_1(B,C,k3_partfun2(B,C,A,E)))
=> k4_finseq_4(B,C,k3_partfun2(B,C,A,E),D) = k4_finseq_4(B,C,E,D) ) ) ) ) ) ).
fof(t40_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,B) )
=> ( D = k9_relat_1(E,A)
<=> ! [F] :
( m1_subset_1(F,B)
=> ( r2_hidden(F,D)
<=> ? [G] :
( m1_subset_1(G,C)
& r2_hidden(G,k4_relset_1(C,B,E))
& r2_hidden(G,A)
& F = k4_finseq_4(C,B,E,G) ) ) ) ) ) ) ) ) ).
fof(t41_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,C)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,B,C) )
=> ( r2_hidden(D,k10_relset_1(B,C,E,A))
<=> ? [F] :
( m1_subset_1(F,B)
& r2_hidden(F,k4_relset_1(B,C,E))
& r2_hidden(F,A)
& D = k4_finseq_4(B,C,E,F) ) ) ) ) ) ) ).
fof(t42_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( r2_hidden(C,k4_relset_1(A,B,D))
=> k9_relat_1(D,k1_tarski(C)) = k1_tarski(k4_finseq_4(A,B,D,C)) ) ) ) ) ) ).
fof(t43_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ( ( r2_hidden(C,k4_relset_1(A,B,E))
& r2_hidden(D,k4_relset_1(A,B,E)) )
=> k9_relat_1(E,k2_tarski(C,D)) = k2_tarski(k4_finseq_4(A,B,E,C),k4_finseq_4(A,B,E,D)) ) ) ) ) ) ) ).
fof(t44_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(C))
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,B) )
=> ( D = k3_funct_2(C,B,E,A)
<=> ! [F] :
( m1_subset_1(F,C)
=> ( r2_hidden(F,D)
<=> ( r2_hidden(F,k4_relset_1(C,B,E))
& r2_hidden(k4_finseq_4(C,B,E,F),A) ) ) ) ) ) ) ) ) ).
fof(t45_partfun2,axiom,
$true ).
fof(t46_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,B,A) )
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,B,A)
& m2_relset_1(D,B,A)
& ! [E] :
( m1_subset_1(E,B)
=> ( r2_hidden(E,k4_relset_1(B,A,C))
=> k8_funct_2(B,A,D,E) = k4_finseq_4(B,A,C,E) ) ) ) ) ) ) ).
fof(t47_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,B,A) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,A) )
=> ( r1_partfun1(C,D)
<=> ! [E] :
( m1_subset_1(E,B)
=> ( r2_hidden(E,k5_subset_1(B,k4_relset_1(B,A,C),k4_relset_1(B,A,D)))
=> k4_finseq_4(B,A,C,E) = k4_finseq_4(B,A,D,E) ) ) ) ) ) ) ) ).
fof(t48_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,B)
=> ( r2_hidden(D,C)
=> k4_finseq_4(A,B,k4_partfun2(A,B,C,E),D) = E ) ) ) ) ) ) ).
fof(t49_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,A) )
=> ( ! [E] :
( m1_subset_1(E,B)
=> ( r2_hidden(E,k4_relset_1(B,A,D))
=> k4_finseq_4(B,A,D,E) = C ) )
=> D = k4_partfun2(B,A,k4_relset_1(B,A,D),C) ) ) ) ) ) ).
fof(t50_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,B,C) )
=> ( r2_hidden(E,k4_relset_1(B,C,F))
=> k1_partfun1(A,B,B,C,k4_partfun2(A,B,D,E),F) = k4_partfun2(A,C,D,k4_finseq_4(B,C,F,E)) ) ) ) ) ) ) ) ).
fof(t51_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_partfun1(k1_partfun2(A,B),A,A)
<=> B = A ) ) ) ).
fof(t52_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,B)
=> ~ ( v1_partfun1(k4_partfun2(A,B,C,D),A,B)
& C = k1_xboole_0 ) ) ) ) ) ).
fof(t53_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ! [D] :
( m1_subset_1(D,A)
=> ( v1_partfun1(k4_partfun2(B,A,C,D),B,A)
<=> C = B ) ) ) ) ) ).
fof(d1_partfun2,axiom,
$true ).
fof(d2_partfun2,axiom,
$true ).
fof(d3_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( r1_partfun2(A,B,C,D)
<=> ? [E] :
( m1_subset_1(E,B)
& ! [F] :
( m1_subset_1(F,A)
=> ( r2_hidden(F,k3_xboole_0(D,k4_relset_1(A,B,C)))
=> k4_finseq_4(A,B,C,F) = E ) ) ) ) ) ) ) ).
fof(t54_partfun2,axiom,
$true ).
fof(t55_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,C,B) )
=> ( r1_partfun2(C,B,D,A)
<=> ! [E] :
( m1_subset_1(E,C)
=> ! [F] :
( m1_subset_1(F,C)
=> ( ( r2_hidden(E,k3_xboole_0(A,k4_relset_1(C,B,D)))
& r2_hidden(F,k3_xboole_0(A,k4_relset_1(C,B,D))) )
=> k4_finseq_4(C,B,D,E) = k4_finseq_4(C,B,D,F) ) ) ) ) ) ) ) ).
fof(t56_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,C) )
=> ( ~ r1_xboole_0(A,k4_relset_1(B,C,D))
=> ( r1_partfun2(B,C,D,A)
<=> ? [E] :
( m1_subset_1(E,C)
& k2_relat_1(k2_partfun1(B,C,D,A)) = k1_tarski(E) ) ) ) ) ) ) ).
fof(t57_partfun2,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,D) )
=> ( ( r1_partfun2(C,D,E,A)
& r1_tarski(B,A) )
=> r1_partfun2(C,D,E,B) ) ) ) ) ).
fof(t58_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,C) )
=> ( r1_xboole_0(A,k4_relset_1(B,C,D))
=> r1_partfun2(B,C,D,A) ) ) ) ) ).
fof(t59_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ( k2_partfun1(A,B,E,C) = k4_partfun2(A,B,k4_relset_1(A,B,k2_partfun1(A,B,E,C)),D)
=> r1_partfun2(A,B,E,C) ) ) ) ) ) ) ).
fof(t60_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,C) )
=> r1_partfun2(B,C,D,k1_tarski(A)) ) ) ) ).
fof(t61_partfun2,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,D) )
=> ( ( r1_partfun2(C,D,E,A)
& r1_partfun2(C,D,E,B) )
=> ( r1_xboole_0(k3_xboole_0(A,B),k4_relset_1(C,D,E))
| r1_partfun2(C,D,E,k2_xboole_0(A,B)) ) ) ) ) ) ).
fof(t62_partfun2,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,D) )
=> ( r1_partfun2(C,D,E,A)
=> r1_partfun2(C,D,k2_partfun1(C,D,E,B),A) ) ) ) ) ).
fof(t63_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,B)
=> r1_partfun2(A,B,k4_partfun2(A,B,C,D),C) ) ) ) ) ).
fof(t64_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,B,A) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,A) )
=> ( r1_tarski(C,D)
<=> ( r1_tarski(k4_relset_1(B,A,C),k4_relset_1(B,A,D))
& ! [E] :
( m1_subset_1(E,B)
=> ( r2_hidden(E,k4_relset_1(B,A,C))
=> k4_finseq_4(B,A,C,E) = k4_finseq_4(B,A,D,E) ) ) ) ) ) ) ) ) ).
fof(t65_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ( ( r2_hidden(C,k4_relset_1(A,B,E))
& D = k4_finseq_4(A,B,E,C) )
<=> r2_hidden(k4_tarski(C,D),E) ) ) ) ) ) ) ).
fof(t66_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,C)
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,A,B) )
=> ! [G] :
( ( v1_funct_1(G)
& m2_relset_1(G,B,C) )
=> ( r2_hidden(k4_tarski(D,E),k1_partfun1(A,B,B,C,F,G))
=> ( r2_hidden(k4_tarski(D,k4_finseq_4(A,B,F,D)),F)
& r2_hidden(k4_tarski(k4_finseq_4(A,B,F,D),E),G) ) ) ) ) ) ) ) ) ) ).
fof(t67_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ( E = k1_tarski(k4_tarski(C,D))
=> k4_finseq_4(A,B,E,C) = D ) ) ) ) ) ) ).
fof(t68_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( k4_relset_1(A,B,D) = k1_tarski(C)
=> D = k1_tarski(k4_tarski(C,k4_finseq_4(A,B,D,C))) ) ) ) ) ) ).
fof(t69_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,A,B) )
=> ( ( D = k2_relset_1(A,B,E,F)
& r2_hidden(C,k4_relset_1(A,B,D)) )
=> ( k4_finseq_4(A,B,D,C) = k4_finseq_4(A,B,E,C)
& k4_finseq_4(A,B,D,C) = k4_finseq_4(A,B,F,C) ) ) ) ) ) ) ) ) ).
fof(t70_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,A,B) )
=> ( ( r2_hidden(C,k4_relset_1(A,B,D))
& E = k1_relset_1(A,B,D,F) )
=> k4_finseq_4(A,B,E,C) = k4_finseq_4(A,B,D,C) ) ) ) ) ) ) ) ).
fof(t71_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,A,B) )
=> ( ( r2_hidden(C,k4_relset_1(A,B,D))
& E = k1_relset_1(A,B,F,D) )
=> k4_finseq_4(A,B,E,C) = k4_finseq_4(A,B,D,C) ) ) ) ) ) ) ) ).
fof(t72_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,A,B) )
=> ~ ( r2_hidden(C,k4_relset_1(A,B,D))
& D = k1_relset_1(A,B,E,F)
& k4_finseq_4(A,B,D,C) != k4_finseq_4(A,B,E,C)
& k4_finseq_4(A,B,D,C) != k4_finseq_4(A,B,F,C) ) ) ) ) ) ) ) ).
fof(t73_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ( ( r2_hidden(D,k4_relset_1(A,B,E))
& r2_hidden(D,C) )
<=> r2_hidden(k4_tarski(D,k4_finseq_4(A,B,E,D)),k2_partfun1(A,B,E,C)) ) ) ) ) ) ) ).
fof(t74_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ( ( r2_hidden(D,k4_relset_1(A,B,E))
& r2_hidden(k4_finseq_4(A,B,E,D),C) )
<=> r2_hidden(k4_tarski(D,k4_finseq_4(A,B,E,D)),k3_partfun2(A,B,C,E)) ) ) ) ) ) ) ).
fof(t75_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ( r2_hidden(D,k3_funct_2(A,B,E,C))
<=> ( r2_hidden(k4_tarski(D,k4_finseq_4(A,B,E,D)),E)
& r2_hidden(k4_finseq_4(A,B,E,D),C) ) ) ) ) ) ) ) ).
fof(t76_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,C,B) )
=> ( r1_partfun2(C,B,D,A)
<=> ? [E] :
( m1_subset_1(E,B)
& ! [F] :
( m1_subset_1(F,C)
=> ( r2_hidden(F,k3_xboole_0(A,k4_relset_1(C,B,D)))
=> k1_funct_1(D,F) = E ) ) ) ) ) ) ) ).
fof(t77_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,C,B) )
=> ( r1_partfun2(C,B,D,A)
<=> ! [E] :
( m1_subset_1(E,C)
=> ! [F] :
( m1_subset_1(F,C)
=> ( ( r2_hidden(E,k3_xboole_0(A,k4_relset_1(C,B,D)))
& r2_hidden(F,k3_xboole_0(A,k4_relset_1(C,B,D))) )
=> k1_funct_1(D,E) = k1_funct_1(D,F) ) ) ) ) ) ) ) ).
fof(t78_partfun2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,B) )
=> ~ ( r2_hidden(D,k9_relat_1(E,A))
& ! [F] :
( m1_subset_1(F,C)
=> ~ ( r2_hidden(F,k4_relset_1(C,B,E))
& r2_hidden(F,A)
& D = k1_funct_1(E,F) ) ) ) ) ) ) ) ).
fof(t79_partfun2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ( v2_funct_1(E)
=> ( ( ( r2_hidden(D,k2_relat_1(E))
& C = k1_funct_1(k2_funct_1(E),D) )
=> ( r2_hidden(C,k4_relset_1(A,B,E))
& D = k1_funct_1(E,C) ) )
& ( ( r2_hidden(C,k4_relset_1(A,B,E))
& D = k1_funct_1(E,C) )
=> ( r2_hidden(D,k2_relat_1(E))
& C = k1_funct_1(k2_funct_1(E),D) ) ) ) ) ) ) ) ) ) ).
fof(s1_partfun2,axiom,
? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s1_partfun2,f2_s1_partfun2)
& ! [B] :
( m1_subset_1(B,f1_s1_partfun2)
=> ( r2_hidden(B,k4_relset_1(f1_s1_partfun2,f2_s1_partfun2,A))
<=> ? [C] :
( m1_subset_1(C,f2_s1_partfun2)
& p1_s1_partfun2(B,C) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s1_partfun2)
=> ( r2_hidden(B,k4_relset_1(f1_s1_partfun2,f2_s1_partfun2,A))
=> p1_s1_partfun2(B,k4_finseq_4(f1_s1_partfun2,f2_s1_partfun2,A,B)) ) ) ) ).
fof(s2_partfun2,axiom,
? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s2_partfun2,f2_s2_partfun2)
& ! [B] :
( m1_subset_1(B,f1_s2_partfun2)
=> ( r2_hidden(B,k4_relset_1(f1_s2_partfun2,f2_s2_partfun2,A))
<=> p1_s2_partfun2(B) ) )
& ! [B] :
( m1_subset_1(B,f1_s2_partfun2)
=> ( r2_hidden(B,k4_relset_1(f1_s2_partfun2,f2_s2_partfun2,A))
=> k4_finseq_4(f1_s2_partfun2,f2_s2_partfun2,A,B) = f3_s2_partfun2(B) ) ) ) ).
fof(s3_partfun2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,f1_s3_partfun2,f2_s3_partfun2) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,f1_s3_partfun2,f2_s3_partfun2) )
=> ( ( k4_relset_1(f1_s3_partfun2,f2_s3_partfun2,A) = f3_s3_partfun2
& ! [C] :
( m1_subset_1(C,f1_s3_partfun2)
=> ( r2_hidden(C,k4_relset_1(f1_s3_partfun2,f2_s3_partfun2,A))
=> k4_finseq_4(f1_s3_partfun2,f2_s3_partfun2,A,C) = f4_s3_partfun2(C) ) )
& k4_relset_1(f1_s3_partfun2,f2_s3_partfun2,B) = f3_s3_partfun2
& ! [C] :
( m1_subset_1(C,f1_s3_partfun2)
=> ( r2_hidden(C,k4_relset_1(f1_s3_partfun2,f2_s3_partfun2,B))
=> k4_finseq_4(f1_s3_partfun2,f2_s3_partfun2,B,C) = f4_s3_partfun2(C) ) ) )
=> A = B ) ) ) ).
fof(dt_k1_partfun2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v1_funct_1(k1_partfun2(A,B))
& m2_relset_1(k1_partfun2(A,B),A,A) ) ) ).
fof(redefinition_k1_partfun2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> k1_partfun2(A,B) = k6_relat_1(B) ) ).
fof(dt_k2_partfun2,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v2_funct_1(C)
& m1_relset_1(C,A,B) )
=> ( v1_funct_1(k2_partfun2(A,B,C))
& m2_relset_1(k2_partfun2(A,B,C),B,A) ) ) ).
fof(redefinition_k2_partfun2,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v2_funct_1(C)
& m1_relset_1(C,A,B) )
=> k2_partfun2(A,B,C) = k2_funct_1(C) ) ).
fof(dt_k3_partfun2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(D)
& m1_relset_1(D,A,B) )
=> ( v1_funct_1(k3_partfun2(A,B,C,D))
& m2_relset_1(k3_partfun2(A,B,C,D),A,B) ) ) ).
fof(redefinition_k3_partfun2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(D)
& m1_relset_1(D,A,B) )
=> k3_partfun2(A,B,C,D) = k8_relat_1(C,D) ) ).
fof(dt_k4_partfun2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k1_zfmisc_1(A))
& m1_subset_1(D,B) )
=> ( v1_funct_1(k4_partfun2(A,B,C,D))
& m2_relset_1(k4_partfun2(A,B,C,D),A,B) ) ) ).
fof(redefinition_k4_partfun2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k1_zfmisc_1(A))
& m1_subset_1(D,B) )
=> k4_partfun2(A,B,C,D) = k2_funcop_1(C,D) ) ).
%------------------------------------------------------------------------------