SET007 Axioms: SET007+17.ax
%------------------------------------------------------------------------------
% File : SET007+17 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Partial 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 : partfun1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 215 ( 86 unt; 0 def)
% Number of atoms : 735 ( 83 equ)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 533 ( 13 ~; 0 |; 276 &)
% ( 15 <=>; 229 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 40 ( 40 usr; 12 con; 0-6 aty)
% Number of variables : 457 ( 443 !; 14 ?)
% 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(rc1_partfun1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_xboole_0(A) ) ).
fof(rc2_partfun1,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C) ) ).
fof(fc1_partfun1,axiom,
! [A,B] : ~ v1_xboole_0(k4_partfun1(A,B)) ).
fof(rc3_partfun1,axiom,
! [A] :
? [B] :
( m1_relset_1(B,A,A)
& v1_relat_1(B)
& v1_relat_2(B)
& v3_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A) ) ).
fof(cc1_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v3_relat_2(A)
& v8_relat_2(A) )
=> ( v1_relat_1(A)
& v1_relat_2(A) ) ) ).
fof(fc2_partfun1,axiom,
! [A] :
( v1_relat_1(k6_relat_1(A))
& v1_funct_1(k6_relat_1(A))
& v1_relat_2(k6_relat_1(A))
& v3_relat_2(k6_relat_1(A))
& v4_relat_2(k6_relat_1(A))
& v8_relat_2(k6_relat_1(A)) ) ).
fof(t1_partfun1,axiom,
! [A,B,C,D,E,F] :
( ( r1_tarski(A,k2_zfmisc_1(B,C))
& r1_tarski(D,k2_zfmisc_1(E,F)) )
=> r1_tarski(k2_xboole_0(A,D),k2_zfmisc_1(k2_xboole_0(B,E),k2_xboole_0(C,F))) ) ).
fof(t2_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ~ ( ! [C] :
( r2_hidden(C,k3_xboole_0(k1_relat_1(A),k1_relat_1(B)))
=> k1_funct_1(A,C) = k1_funct_1(B,C) )
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k2_xboole_0(A,B) != C ) ) ) ) ).
fof(t3_partfun1,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_xboole_0(A,B) = C
=> ! [D] :
( r2_hidden(D,k3_xboole_0(k1_relat_1(A),k1_relat_1(B)))
=> k1_funct_1(A,D) = k1_funct_1(B,D) ) ) ) ) ) ).
fof(t4_partfun1,axiom,
$true ).
fof(t5_partfun1,axiom,
$true ).
fof(t6_partfun1,axiom,
$true ).
fof(t7_partfun1,axiom,
$true ).
fof(t8_partfun1,axiom,
$true ).
fof(t9_partfun1,axiom,
$true ).
fof(t10_partfun1,axiom,
k2_relat_1(k1_xboole_0) = k1_xboole_0 ).
fof(t11_partfun1,axiom,
$true ).
fof(t12_partfun1,axiom,
$true ).
fof(t13_partfun1,axiom,
$true ).
fof(t14_partfun1,axiom,
$true ).
fof(t15_partfun1,axiom,
$true ).
fof(t16_partfun1,axiom,
$true ).
fof(t17_partfun1,axiom,
$true ).
fof(t18_partfun1,axiom,
$true ).
fof(t19_partfun1,axiom,
$true ).
fof(t20_partfun1,axiom,
$true ).
fof(t21_partfun1,axiom,
$true ).
fof(t22_partfun1,axiom,
$true ).
fof(t23_partfun1,axiom,
$true ).
fof(t24_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_funct_1(A)
& m2_relset_1(A,k1_relat_1(A),k2_relat_1(A)) ) ) ).
fof(t25_partfun1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(k2_relat_1(B),A)
=> ( v1_funct_1(B)
& m2_relset_1(B,k1_relat_1(B),A) ) ) ) ).
fof(t26_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ~ ( r2_hidden(C,k2_relat_1(D))
& ! [E] :
( m1_subset_1(E,A)
=> ~ ( r2_hidden(E,k4_relset_1(A,B,D))
& C = k1_funct_1(D,E) ) ) ) ) ).
fof(t27_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( r2_hidden(C,k4_relset_1(A,B,D))
=> r2_hidden(k1_funct_1(D,C),B) ) ) ).
fof(t28_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( r1_tarski(k4_relset_1(A,B,D),C)
=> ( v1_funct_1(D)
& m2_relset_1(D,C,B) ) ) ) ).
fof(t29_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( r1_tarski(k2_relat_1(D),C)
=> ( v1_funct_1(D)
& m2_relset_1(D,A,C) ) ) ) ).
fof(t30_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( r1_tarski(A,C)
=> ( v1_funct_1(D)
& m2_relset_1(D,C,B) ) ) ) ).
fof(t31_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( r1_tarski(B,C)
=> ( v1_funct_1(D)
& m2_relset_1(D,A,C) ) ) ) ).
fof(t32_partfun1,axiom,
! [A,B,C,D,E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ( ( r1_tarski(A,C)
& r1_tarski(B,D) )
=> ( v1_funct_1(E)
& m2_relset_1(E,C,D) ) ) ) ).
fof(t33_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( r1_tarski(C,D)
=> ( v1_funct_1(C)
& m2_relset_1(C,A,B) ) ) ) ) ).
fof(t34_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( ( k4_relset_1(A,B,C) = k4_relset_1(A,B,D)
& ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,k4_relset_1(A,B,C))
=> k1_funct_1(C,E) = k1_funct_1(D,E) ) ) )
=> C = D ) ) ) ).
fof(t35_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,k2_zfmisc_1(A,B),C) )
=> ( ( k4_relset_1(k2_zfmisc_1(A,B),C,D) = k4_relset_1(k2_zfmisc_1(A,B),C,E)
& ! [F,G] :
( r2_hidden(k4_tarski(F,G),k4_relset_1(k2_zfmisc_1(A,B),C,D))
=> k1_funct_1(D,k4_tarski(F,G)) = k1_funct_1(E,k4_tarski(F,G)) ) )
=> D = E ) ) ) ).
fof(t36_partfun1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> k5_relat_1(k6_relat_1(A),C) = C ) ).
fof(t37_partfun1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> k5_relat_1(C,k6_relat_1(B)) = C ) ).
fof(t38_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ( ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( ( r2_hidden(D,k4_relset_1(A,B,C))
& r2_hidden(E,k4_relset_1(A,B,C))
& k1_funct_1(C,D) = k1_funct_1(C,E) )
=> D = E ) ) )
=> v2_funct_1(C) ) ) ).
fof(t39_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ( v2_funct_1(C)
=> ( v1_funct_1(k2_funct_1(C))
& m2_relset_1(k2_funct_1(C),B,A) ) ) ) ).
fof(t40_partfun1,axiom,
$true ).
fof(t41_partfun1,axiom,
$true ).
fof(t42_partfun1,axiom,
$true ).
fof(t43_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( v1_funct_1(k8_relset_1(A,B,D,C))
& m2_relset_1(k8_relset_1(A,B,D,C),C,B) ) ) ).
fof(t44_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( v1_funct_1(k8_relset_1(A,B,D,C))
& m2_relset_1(k8_relset_1(A,B,D,C),A,B) ) ) ).
fof(t45_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( v1_funct_1(k9_relset_1(A,B,C,D))
& m2_relset_1(k9_relset_1(A,B,C,D),A,C) ) ) ).
fof(t46_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( v1_funct_1(k9_relset_1(A,B,C,D))
& m2_relset_1(k9_relset_1(A,B,C,D),A,B) ) ) ).
fof(t47_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( v1_funct_1(k7_relat_1(k8_relat_1(A,C),B))
& m2_relset_1(k7_relat_1(k8_relat_1(A,C),B),B,A) ) ) ).
fof(t48_partfun1,axiom,
$true ).
fof(t49_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ~ ( r2_hidden(C,k9_relat_1(D,A))
& ! [E] :
( m1_subset_1(E,A)
=> ~ ( r2_hidden(E,k4_relset_1(A,B,D))
& C = k1_funct_1(D,E) ) ) ) ) ).
fof(t50_partfun1,axiom,
$true ).
fof(t51_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> k9_relat_1(C,A) = k2_relat_1(C) ) ).
fof(t52_partfun1,axiom,
$true ).
fof(t53_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> k10_relat_1(C,B) = k4_relset_1(A,B,C) ) ).
fof(t54_partfun1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_xboole_0,A) )
=> ( k4_relset_1(k1_xboole_0,A,B) = k1_xboole_0
& k2_relat_1(B) = k1_xboole_0 ) ) ).
fof(t55_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k1_relat_1(C) = k1_xboole_0
=> ( v1_funct_1(C)
& m2_relset_1(C,A,B) ) ) ) ).
fof(t56_partfun1,axiom,
! [A,B] :
( v1_funct_1(k1_xboole_0)
& m2_relset_1(k1_xboole_0,A,B) ) ).
fof(t57_partfun1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_xboole_0,A) )
=> B = k1_xboole_0 ) ).
fof(t58_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_xboole_0,A) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_xboole_0,B) )
=> C = D ) ) ).
fof(t59_partfun1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_xboole_0,A) )
=> v2_funct_1(B) ) ).
fof(t60_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_xboole_0,A) )
=> k9_relat_1(C,B) = k1_xboole_0 ) ).
fof(t61_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_xboole_0,A) )
=> k10_relat_1(C,B) = k1_xboole_0 ) ).
fof(t62_partfun1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_xboole_0) )
=> ( k4_relset_1(A,k1_xboole_0,B) = k1_xboole_0
& k2_relat_1(B) = k1_xboole_0 ) ) ).
fof(t63_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k2_relat_1(C) = k1_xboole_0
=> ( v1_funct_1(C)
& m2_relset_1(C,A,B) ) ) ) ).
fof(t64_partfun1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_xboole_0) )
=> B = k1_xboole_0 ) ).
fof(t65_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_xboole_0) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,k1_xboole_0) )
=> C = D ) ) ).
fof(t66_partfun1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_xboole_0) )
=> v2_funct_1(B) ) ).
fof(t67_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_xboole_0) )
=> k9_relat_1(C,B) = k1_xboole_0 ) ).
fof(t68_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_xboole_0) )
=> k10_relat_1(C,B) = k1_xboole_0 ) ).
fof(t69_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_tarski(A),B) )
=> r1_tarski(k2_relat_1(C),k1_tarski(k1_funct_1(C,A))) ) ).
fof(t70_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_tarski(A),B) )
=> v2_funct_1(C) ) ).
fof(t71_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_tarski(A),B) )
=> r1_tarski(k9_relat_1(D,C),k1_tarski(k1_funct_1(D,A))) ) ).
fof(t72_partfun1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( k1_relat_1(D) = k1_tarski(A)
& r2_hidden(A,B)
& r2_hidden(k1_funct_1(D,A),C) )
=> ( v1_funct_1(D)
& m2_relset_1(D,B,C) ) ) ) ).
fof(t73_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k1_tarski(B)) )
=> ( r2_hidden(C,k4_relset_1(A,k1_tarski(B),D))
=> k1_funct_1(D,C) = B ) ) ).
fof(t74_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_tarski(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k1_tarski(B)) )
=> ( k4_relset_1(A,k1_tarski(B),C) = k4_relset_1(A,k1_tarski(B),D)
=> C = D ) ) ) ).
fof(d1_partfun1,axiom,
$true ).
fof(d2_partfun1,axiom,
$true ).
fof(d3_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] : k3_partfun1(A,B,C) = k7_relat_1(k8_relat_1(C,A),B) ) ).
fof(t75_partfun1,axiom,
$true ).
fof(t76_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> r1_tarski(k3_partfun1(C,A,B),C) ) ).
fof(t77_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_tarski(k4_relset_1(A,B,k3_partfun1(C,A,B)),k1_relat_1(C))
& r1_tarski(k2_relat_1(k3_partfun1(C,A,B)),k2_relat_1(C)) ) ) ).
fof(t78_partfun1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(A,k4_relset_1(B,C,k3_partfun1(D,B,C)))
<=> ( r2_hidden(A,k1_relat_1(D))
& r2_hidden(A,B)
& r2_hidden(k1_funct_1(D,A),C) ) ) ) ).
fof(t79_partfun1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r2_hidden(A,k1_relat_1(D))
& r2_hidden(A,B)
& r2_hidden(k1_funct_1(D,A),C) )
=> k1_funct_1(k3_partfun1(D,B,C),A) = k1_funct_1(D,A) ) ) ).
fof(t80_partfun1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(A,k4_relset_1(B,C,k3_partfun1(D,B,C)))
=> k1_funct_1(k3_partfun1(D,B,C),A) = k1_funct_1(D,A) ) ) ).
fof(t81_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r1_tarski(C,D)
=> r1_tarski(k3_partfun1(C,A,B),k3_partfun1(D,A,B)) ) ) ) ).
fof(t82_partfun1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r1_tarski(A,B)
=> r1_tarski(k3_partfun1(D,A,C),k3_partfun1(D,B,C)) ) ) ).
fof(t83_partfun1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r1_tarski(A,B)
=> r1_tarski(k3_partfun1(D,C,A),k3_partfun1(D,C,B)) ) ) ).
fof(t84_partfun1,axiom,
! [A,B,C,D,E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> r1_tarski(k3_partfun1(E,A,C),k3_partfun1(E,B,D)) ) ) ).
fof(t85_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(k1_relat_1(C),A)
& r1_tarski(k2_relat_1(C),B) )
=> C = k3_partfun1(C,A,B) ) ) ).
fof(t86_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> A = k3_partfun1(A,k1_relat_1(A),k2_relat_1(A)) ) ).
fof(t87_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> k3_partfun1(C,A,B) = C ) ).
fof(t88_partfun1,axiom,
$true ).
fof(t89_partfun1,axiom,
$true ).
fof(t90_partfun1,axiom,
$true ).
fof(t91_partfun1,axiom,
! [A,B] : k3_partfun1(k1_xboole_0,A,B) = k1_xboole_0 ).
fof(t92_partfun1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> r1_tarski(k1_partfun1(C,A,A,B,k3_partfun1(D,C,A),k3_partfun1(E,A,B)),k3_partfun1(k5_relat_1(D,E),C,B)) ) ) ).
fof(t93_partfun1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( r1_tarski(k3_xboole_0(k2_relat_1(D),k1_relat_1(E)),A)
=> k1_partfun1(C,A,A,B,k3_partfun1(D,C,A),k3_partfun1(E,A,B)) = k3_partfun1(k5_relat_1(D,E),C,B) ) ) ) ).
fof(t94_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( v2_funct_1(C)
=> v2_funct_1(k3_partfun1(C,A,B)) ) ) ).
fof(t95_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( v2_funct_1(C)
=> k2_funct_1(k3_partfun1(C,A,B)) = k3_partfun1(k2_funct_1(C),B,A) ) ) ).
fof(t96_partfun1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> k2_partfun1(A,B,k3_partfun1(D,A,B),C) = k3_partfun1(D,k3_xboole_0(A,C),B) ) ).
fof(t97_partfun1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> k9_relset_1(B,C,A,k3_partfun1(D,B,C)) = k3_partfun1(D,B,k3_xboole_0(A,C)) ) ).
fof(d4_partfun1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ( v1_partfun1(C,A,B)
<=> k4_relset_1(A,B,C) = A ) ) ).
fof(t98_partfun1,axiom,
$true ).
fof(t99_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ( ( v1_partfun1(C,A,B)
& B = k1_xboole_0 )
=> A = k1_xboole_0 ) ) ).
fof(t100_partfun1,axiom,
$true ).
fof(t101_partfun1,axiom,
$true ).
fof(t102_partfun1,axiom,
$true ).
fof(t103_partfun1,axiom,
$true ).
fof(t104_partfun1,axiom,
$true ).
fof(t105_partfun1,axiom,
$true ).
fof(t106_partfun1,axiom,
$true ).
fof(t107_partfun1,axiom,
$true ).
fof(t108_partfun1,axiom,
$true ).
fof(t109_partfun1,axiom,
$true ).
fof(t110_partfun1,axiom,
$true ).
fof(t111_partfun1,axiom,
$true ).
fof(t112_partfun1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_xboole_0,A) )
=> v1_partfun1(B,k1_xboole_0,A) ) ).
fof(t113_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( v1_partfun1(k3_partfun1(C,A,B),A,B)
=> r1_tarski(A,k1_relat_1(C)) ) ) ).
fof(t114_partfun1,axiom,
! [A,B] :
( v1_partfun1(k3_partfun1(k1_xboole_0,A,B),A,B)
=> A = k1_xboole_0 ) ).
fof(t115_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(A,k1_relat_1(C))
& r1_tarski(k2_relat_1(C),B) )
=> v1_partfun1(k3_partfun1(C,A,B),A,B) ) ) ).
fof(t116_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( v1_partfun1(k3_partfun1(C,A,B),A,B)
=> r1_tarski(k9_relat_1(C,A),B) ) ) ).
fof(t117_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(A,k1_relat_1(C))
& r1_tarski(k9_relat_1(C,A),B) )
=> v1_partfun1(k3_partfun1(C,A,B),A,B) ) ) ).
fof(d5_partfun1,axiom,
! [A,B,C] :
( C = k4_partfun1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( v1_relat_1(E)
& v1_funct_1(E)
& D = E
& r1_tarski(k1_relat_1(E),A)
& r1_tarski(k2_relat_1(E),B) ) ) ) ).
fof(t118_partfun1,axiom,
$true ).
fof(t119_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> r2_hidden(C,k4_partfun1(A,B)) ) ).
fof(t120_partfun1,axiom,
! [A,B,C] :
( r2_hidden(C,k4_partfun1(A,B))
=> ( v1_funct_1(C)
& m2_relset_1(C,A,B) ) ) ).
fof(t121_partfun1,axiom,
! [A,B,C] :
( m1_subset_1(C,k4_partfun1(A,B))
=> ( v1_funct_1(C)
& m2_relset_1(C,A,B) ) ) ).
fof(t122_partfun1,axiom,
! [A] : k4_partfun1(k1_xboole_0,A) = k1_tarski(k1_xboole_0) ).
fof(t123_partfun1,axiom,
! [A] : k4_partfun1(A,k1_xboole_0) = k1_tarski(k1_xboole_0) ).
fof(t124_partfun1,axiom,
$true ).
fof(t125_partfun1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> r1_tarski(k4_partfun1(A,C),k4_partfun1(B,C)) ) ).
fof(t126_partfun1,axiom,
! [A,B] : r1_tarski(k4_partfun1(k1_xboole_0,A),k4_partfun1(B,A)) ).
fof(t127_partfun1,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> r1_tarski(k4_partfun1(C,A),k4_partfun1(C,B)) ) ).
fof(t128_partfun1,axiom,
! [A,B,C,D] :
( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> r1_tarski(k4_partfun1(A,C),k4_partfun1(B,D)) ) ).
fof(d6_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_partfun1(A,B)
<=> ! [C] :
( r2_hidden(C,k3_xboole_0(k1_relat_1(A),k1_relat_1(B)))
=> k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) ) ).
fof(t129_partfun1,axiom,
$true ).
fof(t130_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_partfun1(A,B)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& k2_xboole_0(A,B) = C ) ) ) ) ).
fof(t131_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_partfun1(A,B)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& r1_tarski(A,C)
& r1_tarski(B,C) ) ) ) ) ).
fof(t132_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(k1_relat_1(A),k1_relat_1(B))
=> ( r1_partfun1(A,B)
<=> ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) ) ) ).
fof(t133_partfun1,axiom,
$true ).
fof(t134_partfun1,axiom,
$true ).
fof(t135_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(A,B)
=> r1_partfun1(A,B) ) ) ) ).
fof(t136_partfun1,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)
& r1_partfun1(A,B) )
=> A = B ) ) ) ).
fof(t137_partfun1,axiom,
$true ).
fof(t138_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
=> r1_partfun1(A,B) ) ) ) ).
fof(t139_partfun1,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) )
=> ( ( r1_tarski(A,C)
& r1_tarski(B,C) )
=> r1_partfun1(A,B) ) ) ) ) ).
fof(t140_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r1_partfun1(C,E)
& r1_tarski(D,C) )
=> r1_partfun1(D,E) ) ) ) ) ).
fof(t141_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> r1_partfun1(k1_xboole_0,A) ) ).
fof(t142_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> r1_partfun1(k3_partfun1(k1_xboole_0,A,B),C) ) ).
fof(t143_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_tarski(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k1_tarski(B)) )
=> r1_partfun1(C,D) ) ) ).
fof(t144_partfun1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_partfun1(k7_relat_1(B,A),B) ) ).
fof(t145_partfun1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_partfun1(k8_relat_1(A,B),B) ) ).
fof(t146_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> r1_partfun1(k7_relat_1(k8_relat_1(A,C),B),C) ) ).
fof(t147_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> r1_partfun1(k3_partfun1(C,A,B),C) ) ).
fof(t148_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( ( v1_partfun1(C,A,B)
& v1_partfun1(D,A,B)
& r1_partfun1(C,D) )
=> C = D ) ) ) ).
fof(t149_partfun1,axiom,
$true ).
fof(t150_partfun1,axiom,
$true ).
fof(t151_partfun1,axiom,
$true ).
fof(t152_partfun1,axiom,
$true ).
fof(t153_partfun1,axiom,
$true ).
fof(t154_partfun1,axiom,
$true ).
fof(t155_partfun1,axiom,
$true ).
fof(t156_partfun1,axiom,
$true ).
fof(t157_partfun1,axiom,
$true ).
fof(t158_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ( ( r1_partfun1(C,E)
& r1_partfun1(D,E)
& v1_partfun1(E,A,B) )
=> r1_partfun1(C,D) ) ) ) ) ).
fof(t159_partfun1,axiom,
$true ).
fof(t160_partfun1,axiom,
$true ).
fof(t161_partfun1,axiom,
$true ).
fof(t162_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ~ ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
& r1_partfun1(C,D)
& ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,B) )
=> ~ ( v1_partfun1(E,A,B)
& r1_partfun1(C,E)
& r1_partfun1(D,E) ) ) ) ) ) ).
fof(d7_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( D = k5_partfun1(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( v1_funct_1(F)
& m2_relset_1(F,A,B)
& F = E
& v1_partfun1(F,A,B)
& r1_partfun1(C,F) ) ) ) ) ).
fof(t163_partfun1,axiom,
$true ).
fof(t164_partfun1,axiom,
$true ).
fof(t165_partfun1,axiom,
$true ).
fof(t166_partfun1,axiom,
$true ).
fof(t167_partfun1,axiom,
$true ).
fof(t168_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( r2_hidden(D,k5_partfun1(A,B,C))
=> ( v1_funct_1(D)
& m2_relset_1(D,A,B) ) ) ) ).
fof(t169_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( r2_hidden(D,k5_partfun1(A,B,C))
=> v1_partfun1(D,A,B) ) ) ) ).
fof(t170_partfun1,axiom,
$true ).
fof(t171_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(D,k5_partfun1(A,B,C))
=> r1_partfun1(C,D) ) ) ) ).
fof(t172_partfun1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_xboole_0) )
=> ( A != k1_xboole_0
=> k5_partfun1(A,k1_xboole_0,B) = k1_xboole_0 ) ) ).
fof(t173_partfun1,axiom,
$true ).
fof(t174_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ( v1_partfun1(C,A,B)
<=> k5_partfun1(A,B,C) = k1_tarski(C) ) ) ).
fof(t175_partfun1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_xboole_0,A) )
=> k5_partfun1(k1_xboole_0,A,B) = k1_tarski(B) ) ).
fof(t176_partfun1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_xboole_0,A) )
=> k5_partfun1(k1_xboole_0,A,B) = k1_tarski(k1_xboole_0) ) ).
fof(t177_partfun1,axiom,
$true ).
fof(t178_partfun1,axiom,
$true ).
fof(t179_partfun1,axiom,
$true ).
fof(t180_partfun1,axiom,
$true ).
fof(t181_partfun1,axiom,
$true ).
fof(t182_partfun1,axiom,
$true ).
fof(t183_partfun1,axiom,
$true ).
fof(t184_partfun1,axiom,
$true ).
fof(t185_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( ~ r1_xboole_0(k5_partfun1(A,B,C),k5_partfun1(A,B,D))
=> r1_partfun1(C,D) ) ) ) ).
fof(t186_partfun1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ~ ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
& r1_partfun1(C,D)
& r1_xboole_0(k5_partfun1(A,B,C),k5_partfun1(A,B,D)) ) ) ) ).
fof(s1_partfun1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s1_partfun1
& ! [B] :
( r2_hidden(B,f1_s1_partfun1)
=> ( ( p1_s1_partfun1(B)
=> k1_funct_1(A,B) = f2_s1_partfun1(B) )
& ( ~ p1_s1_partfun1(B)
=> k1_funct_1(A,B) = f3_s1_partfun1(B) ) ) ) ) ).
fof(s2_partfun1,axiom,
( ( ! [A,B] :
( ( r2_hidden(A,f1_s2_partfun1)
& p1_s2_partfun1(A,B) )
=> r2_hidden(B,f2_s2_partfun1) )
& ! [A,B,C] :
( ( r2_hidden(A,f1_s2_partfun1)
& p1_s2_partfun1(A,B)
& p1_s2_partfun1(A,C) )
=> B = C ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s2_partfun1,f2_s2_partfun1)
& ! [B] :
( r2_hidden(B,k4_relset_1(f1_s2_partfun1,f2_s2_partfun1,A))
<=> ( r2_hidden(B,f1_s2_partfun1)
& ? [C] : p1_s2_partfun1(B,C) ) )
& ! [B] :
( r2_hidden(B,k4_relset_1(f1_s2_partfun1,f2_s2_partfun1,A))
=> p1_s2_partfun1(B,k1_funct_1(A,B)) ) ) ) ).
fof(s3_partfun1,axiom,
( ! [A] :
( p1_s3_partfun1(A)
=> r2_hidden(f3_s3_partfun1(A),f2_s3_partfun1) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s3_partfun1,f2_s3_partfun1)
& ! [B] :
( r2_hidden(B,k4_relset_1(f1_s3_partfun1,f2_s3_partfun1,A))
<=> ( r2_hidden(B,f1_s3_partfun1)
& p1_s3_partfun1(B) ) )
& ! [B] :
( r2_hidden(B,k4_relset_1(f1_s3_partfun1,f2_s3_partfun1,A))
=> k1_funct_1(A,B) = f3_s3_partfun1(B) ) ) ) ).
fof(s4_partfun1,axiom,
( ( ! [A,B,C] :
( ( r2_hidden(A,f1_s4_partfun1)
& r2_hidden(B,f2_s4_partfun1)
& p1_s4_partfun1(A,B,C) )
=> r2_hidden(C,f3_s4_partfun1) )
& ! [A,B,C,D] :
( ( r2_hidden(A,f1_s4_partfun1)
& r2_hidden(B,f2_s4_partfun1)
& p1_s4_partfun1(A,B,C)
& p1_s4_partfun1(A,B,D) )
=> C = D ) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,k2_zfmisc_1(f1_s4_partfun1,f2_s4_partfun1),f3_s4_partfun1)
& ! [B,C] :
( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s4_partfun1,f2_s4_partfun1),f3_s4_partfun1,A))
<=> ( r2_hidden(B,f1_s4_partfun1)
& r2_hidden(C,f2_s4_partfun1)
& ? [D] : p1_s4_partfun1(B,C,D) ) )
& ! [B,C] :
( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s4_partfun1,f2_s4_partfun1),f3_s4_partfun1,A))
=> p1_s4_partfun1(B,C,k1_funct_1(A,k4_tarski(B,C))) ) ) ) ).
fof(s5_partfun1,axiom,
( ! [A,B] :
( p1_s5_partfun1(A,B)
=> r2_hidden(f4_s5_partfun1(A,B),f3_s5_partfun1) )
=> ? [A] :
( v1_funct_1(A)
& m2_relset_1(A,k2_zfmisc_1(f1_s5_partfun1,f2_s5_partfun1),f3_s5_partfun1)
& ! [B,C] :
( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s5_partfun1,f2_s5_partfun1),f3_s5_partfun1,A))
<=> ( r2_hidden(B,f1_s5_partfun1)
& r2_hidden(C,f2_s5_partfun1)
& p1_s5_partfun1(B,C) ) )
& ! [B,C] :
( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s5_partfun1,f2_s5_partfun1),f3_s5_partfun1,A))
=> k1_funct_1(A,k4_tarski(B,C)) = f4_s5_partfun1(B,C) ) ) ) ).
fof(symmetry_r1_partfun1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_partfun1(A,B)
=> r1_partfun1(B,A) ) ) ).
fof(reflexivity_r1_partfun1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> r1_partfun1(A,A) ) ).
fof(dt_k1_partfun1,axiom,
! [A,B,C,D,E,F] :
( ( v1_funct_1(E)
& m1_relset_1(E,A,B)
& v1_funct_1(F)
& m1_relset_1(F,C,D) )
=> ( v1_funct_1(k1_partfun1(A,B,C,D,E,F))
& m2_relset_1(k1_partfun1(A,B,C,D,E,F),A,D) ) ) ).
fof(redefinition_k1_partfun1,axiom,
! [A,B,C,D,E,F] :
( ( v1_funct_1(E)
& m1_relset_1(E,A,B)
& v1_funct_1(F)
& m1_relset_1(F,C,D) )
=> k1_partfun1(A,B,C,D,E,F) = k5_relat_1(E,F) ) ).
fof(dt_k2_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> ( v1_funct_1(k2_partfun1(A,B,C,D))
& m2_relset_1(k2_partfun1(A,B,C,D),A,B) ) ) ).
fof(redefinition_k2_partfun1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> k2_partfun1(A,B,C,D) = k7_relat_1(C,D) ) ).
fof(dt_k3_partfun1,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_funct_1(k3_partfun1(A,B,C))
& m2_relset_1(k3_partfun1(A,B,C),B,C) ) ) ).
fof(dt_k4_partfun1,axiom,
$true ).
fof(dt_k5_partfun1,axiom,
$true ).
fof(dt_k6_partfun1,axiom,
! [A] :
( v1_partfun1(k6_partfun1(A),A,A)
& m2_relset_1(k6_partfun1(A),A,A) ) ).
fof(redefinition_k6_partfun1,axiom,
! [A] : k6_partfun1(A) = k6_relat_1(A) ).
%------------------------------------------------------------------------------