SET007 Axioms: SET007+11.ax
%------------------------------------------------------------------------------
% File : SET007+11 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Functions and Their Basic Properties
% 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 196 ( 78 unt; 0 def)
% Number of atoms : 705 ( 145 equ)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 526 ( 17 ~; 0 |; 258 &)
% ( 28 <=>; 223 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 15 ( 13 usr; 1 prp; 0-2 aty)
% Number of functors : 24 ( 24 usr; 5 con; 0-2 aty)
% Number of variables : 308 ( 294 !; 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_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_funct_1(A) ) ).
fof(fc1_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k5_relat_1(A,B))
& v1_funct_1(k5_relat_1(A,B)) ) ) ).
fof(fc2_funct_1,axiom,
! [A] :
( v1_relat_1(k6_relat_1(A))
& v1_funct_1(k6_relat_1(A)) ) ).
fof(rc2_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) ) ).
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(rc3_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) ) ).
fof(fc3_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) )
=> ( v1_relat_1(k4_relat_1(A))
& v1_funct_1(k4_relat_1(A)) ) ) ).
fof(fc4_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k7_relat_1(A,B))
& v1_funct_1(k7_relat_1(A,B)) ) ) ).
fof(fc5_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k8_relat_1(A,B))
& v1_funct_1(k8_relat_1(A,B)) ) ) ).
fof(rc4_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v3_relat_1(A)
& v1_funct_1(A) ) ).
fof(rc5_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A) ) ).
fof(fc6_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A) )
=> v1_setfam_1(k2_relat_1(A)) ) ).
fof(d1_funct_1,axiom,
! [A] :
( v1_funct_1(A)
<=> ! [B,C,D] :
( ( r2_hidden(k4_tarski(B,C),A)
& r2_hidden(k4_tarski(B,D),A) )
=> C = D ) ) ).
fof(t1_funct_1,axiom,
$true ).
fof(t2_funct_1,axiom,
! [A] :
( ( ! [B] :
~ ( r2_hidden(B,A)
& ! [C,D] : k4_tarski(C,D) != B )
& ! [B,C,D] :
( ( r2_hidden(k4_tarski(B,C),A)
& r2_hidden(k4_tarski(B,D),A) )
=> C = D ) )
=> ( v1_relat_1(A)
& v1_funct_1(A) ) ) ).
fof(d2_funct_1,axiom,
$true ).
fof(d3_funct_1,axiom,
$true ).
fof(d4_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( ( r2_hidden(B,k1_relat_1(A))
=> ( C = k1_funct_1(A,B)
<=> r2_hidden(k4_tarski(B,C),A) ) )
& ( ~ r2_hidden(B,k1_relat_1(A))
=> ( C = k1_funct_1(A,B)
<=> C = k1_xboole_0 ) ) ) ) ).
fof(t3_funct_1,axiom,
$true ).
fof(t4_funct_1,axiom,
$true ).
fof(t5_funct_1,axiom,
$true ).
fof(t6_funct_1,axiom,
$true ).
fof(t7_funct_1,axiom,
$true ).
fof(t8_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(k4_tarski(A,B),C)
<=> ( r2_hidden(A,k1_relat_1(C))
& B = k1_funct_1(C,A) ) ) ) ).
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 ) ) ) ).
fof(d5_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( B = k2_relat_1(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] :
( r2_hidden(D,k1_relat_1(A))
& C = k1_funct_1(A,D) ) ) ) ) ).
fof(t10_funct_1,axiom,
$true ).
fof(t11_funct_1,axiom,
$true ).
fof(t12_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k1_relat_1(B))
=> r2_hidden(k1_funct_1(B,A),k2_relat_1(B)) ) ) ).
fof(t13_funct_1,axiom,
$true ).
fof(t14_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( k1_relat_1(B) = k1_tarski(A)
=> k2_relat_1(B) = k1_tarski(k1_funct_1(B,A)) ) ) ).
fof(t15_funct_1,axiom,
! [A] :
( A != k1_xboole_0
=> ! [B] :
? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& k1_relat_1(C) = A
& k2_relat_1(C) = k1_tarski(B) ) ) ).
fof(t16_funct_1,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 )
=> B = C ) ) )
=> A = k1_xboole_0 ) ).
fof(t17_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( k1_relat_1(B) = k1_relat_1(C)
& k2_relat_1(B) = k1_tarski(A)
& k2_relat_1(C) = k1_tarski(A) )
=> B = C ) ) ) ).
fof(t18_funct_1,axiom,
! [A,B] :
~ ( ~ ( A = k1_xboole_0
& B != k1_xboole_0 )
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ~ ( B = k1_relat_1(C)
& r1_tarski(k2_relat_1(C),A) ) ) ) ).
fof(t19_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ! [C] :
~ ( r2_hidden(C,A)
& ! [D] :
~ ( r2_hidden(D,k1_relat_1(B))
& C = k1_funct_1(B,D) ) )
=> r1_tarski(A,k2_relat_1(B)) ) ) ).
fof(t20_funct_1,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) )
=> ( ( ! [D] :
( r2_hidden(D,k1_relat_1(C))
<=> ( r2_hidden(D,k1_relat_1(A))
& r2_hidden(k1_funct_1(A,D),k1_relat_1(B)) ) )
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k1_funct_1(B,k1_funct_1(A,D)) ) )
=> C = k5_relat_1(A,B) ) ) ) ) ).
fof(t21_funct_1,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(k5_relat_1(C,B)))
<=> ( r2_hidden(A,k1_relat_1(C))
& r2_hidden(k1_funct_1(C,A),k1_relat_1(B)) ) ) ) ) ).
fof(t22_funct_1,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(k5_relat_1(C,B)))
=> k1_funct_1(k5_relat_1(C,B),A) = k1_funct_1(B,k1_funct_1(C,A)) ) ) ) ).
fof(t23_funct_1,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))
=> k1_funct_1(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) ).
fof(t24_funct_1,axiom,
$true ).
fof(t25_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(A,k2_relat_1(k5_relat_1(C,B)))
=> r2_hidden(A,k2_relat_1(B)) ) ) ) ).
fof(t26_funct_1,axiom,
$true ).
fof(t27_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( k1_relat_1(k5_relat_1(B,A)) = k1_relat_1(B)
=> r1_tarski(k2_relat_1(B),k1_relat_1(A)) ) ) ) ).
fof(t28_funct_1,axiom,
$true ).
fof(t29_funct_1,axiom,
$true ).
fof(t30_funct_1,axiom,
$true ).
fof(t31_funct_1,axiom,
$true ).
fof(t32_funct_1,axiom,
$true ).
fof(t33_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( r1_tarski(k2_relat_1(B),A)
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( k1_relat_1(C) = A
& k1_relat_1(D) = A
& k5_relat_1(B,C) = k5_relat_1(B,D) )
=> C = D ) ) ) )
=> A = k2_relat_1(B) ) ) ).
fof(t34_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k6_relat_1(A)
<=> ( k1_relat_1(B) = A
& ! [C] :
( r2_hidden(C,A)
=> k1_funct_1(B,C) = C ) ) ) ) ).
fof(t35_funct_1,axiom,
! [A,B] :
( r2_hidden(B,A)
=> k1_funct_1(k6_relat_1(A),B) = B ) ).
fof(t36_funct_1,axiom,
$true ).
fof(t37_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k1_relat_1(k5_relat_1(k6_relat_1(A),B)) = k3_xboole_0(k1_relat_1(B),A) ) ).
fof(t38_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(B,k3_xboole_0(k1_relat_1(C),A))
=> k1_funct_1(C,B) = k1_funct_1(k5_relat_1(k6_relat_1(A),C),B) ) ) ).
fof(t39_funct_1,axiom,
$true ).
fof(t40_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(B,k1_relat_1(k5_relat_1(C,k6_relat_1(A))))
<=> ( r2_hidden(B,k1_relat_1(C))
& r2_hidden(k1_funct_1(C,B),A) ) ) ) ).
fof(t41_funct_1,axiom,
$true ).
fof(t42_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( k5_relat_1(k6_relat_1(k1_relat_1(A)),A) = A
& k5_relat_1(A,k6_relat_1(k2_relat_1(A))) = A ) ) ).
fof(t43_funct_1,axiom,
! [A,B] : k5_relat_1(k6_relat_1(B),k6_relat_1(A)) = k6_relat_1(k3_xboole_0(A,B)) ).
fof(t44_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k2_relat_1(A) = k1_relat_1(B)
& k5_relat_1(A,B) = A )
=> B = k6_relat_1(k1_relat_1(B)) ) ) ) ).
fof(d6_funct_1,axiom,
$true ).
fof(d7_funct_1,axiom,
$true ).
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(t45_funct_1,axiom,
$true ).
fof(t46_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_funct_1(A)
& v2_funct_1(B) )
=> v2_funct_1(k5_relat_1(A,B)) ) ) ) ).
fof(t47_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_funct_1(k5_relat_1(B,A))
& r1_tarski(k2_relat_1(B),k1_relat_1(A)) )
=> v2_funct_1(B) ) ) ) ).
fof(t48_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_funct_1(k5_relat_1(B,A))
& k2_relat_1(B) = k1_relat_1(A) )
=> ( v2_funct_1(B)
& v2_funct_1(A) ) ) ) ) ).
fof(t49_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
<=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(k2_relat_1(B),k1_relat_1(A))
& r1_tarski(k2_relat_1(C),k1_relat_1(A))
& k1_relat_1(B) = k1_relat_1(C)
& k5_relat_1(B,A) = k5_relat_1(C,A) )
=> B = C ) ) ) ) ) ).
fof(t50_funct_1,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
& r1_tarski(k2_relat_1(C),A)
& v2_funct_1(B)
& k5_relat_1(C,B) = B )
=> C = k6_relat_1(A) ) ) ) ).
fof(t51_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k2_relat_1(k5_relat_1(B,A)) = k2_relat_1(A)
& v2_funct_1(A) )
=> r1_tarski(k1_relat_1(A),k2_relat_1(B)) ) ) ) ).
fof(t52_funct_1,axiom,
! [A] : v2_funct_1(k6_relat_1(A)) ).
fof(t53_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ? [B] :
( v1_relat_1(B)
& v1_funct_1(B)
& k5_relat_1(A,B) = k6_relat_1(k1_relat_1(A)) )
=> v2_funct_1(A) ) ) ).
fof(d9_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
=> k2_funct_1(A) = k4_relat_1(A) ) ) ).
fof(t54_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k2_funct_1(A)
<=> ( k1_relat_1(B) = k2_relat_1(A)
& ! [C,D] :
( ( ( r2_hidden(C,k2_relat_1(A))
& D = k1_funct_1(B,C) )
=> ( r2_hidden(D,k1_relat_1(A))
& C = k1_funct_1(A,D) ) )
& ( ( r2_hidden(D,k1_relat_1(A))
& C = k1_funct_1(A,D) )
=> ( r2_hidden(C,k2_relat_1(A))
& D = k1_funct_1(B,C) ) ) ) ) ) ) ) ) ).
fof(t55_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
=> ( k2_relat_1(A) = k1_relat_1(k2_funct_1(A))
& k1_relat_1(A) = k2_relat_1(k2_funct_1(A)) ) ) ) ).
fof(t56_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_funct_1(B)
& r2_hidden(A,k1_relat_1(B)) )
=> ( A = k1_funct_1(k2_funct_1(B),k1_funct_1(B,A))
& A = k1_funct_1(k5_relat_1(B,k2_funct_1(B)),A) ) ) ) ).
fof(t57_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_funct_1(B)
& r2_hidden(A,k2_relat_1(B)) )
=> ( A = k1_funct_1(B,k1_funct_1(k2_funct_1(B),A))
& A = k1_funct_1(k5_relat_1(k2_funct_1(B),B),A) ) ) ) ).
fof(t58_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
=> ( k1_relat_1(k5_relat_1(A,k2_funct_1(A))) = k1_relat_1(A)
& k2_relat_1(k5_relat_1(A,k2_funct_1(A))) = k1_relat_1(A) ) ) ) ).
fof(t59_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
=> ( k1_relat_1(k5_relat_1(k2_funct_1(A),A)) = k2_relat_1(A)
& k2_relat_1(k5_relat_1(k2_funct_1(A),A)) = k2_relat_1(A) ) ) ) ).
fof(t60_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_funct_1(A)
& k1_relat_1(A) = k2_relat_1(B)
& k2_relat_1(A) = k1_relat_1(B)
& ! [C,D] :
( ( r2_hidden(C,k1_relat_1(A))
& r2_hidden(D,k1_relat_1(B)) )
=> ( k1_funct_1(A,C) = D
<=> k1_funct_1(B,D) = C ) ) )
=> B = k2_funct_1(A) ) ) ) ).
fof(t61_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
=> ( k5_relat_1(A,k2_funct_1(A)) = k6_relat_1(k1_relat_1(A))
& k5_relat_1(k2_funct_1(A),A) = k6_relat_1(k2_relat_1(A)) ) ) ) ).
fof(t62_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
=> v2_funct_1(k2_funct_1(A)) ) ) ).
fof(t63_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_funct_1(A)
& k2_relat_1(A) = k1_relat_1(B)
& k5_relat_1(A,B) = k6_relat_1(k1_relat_1(A)) )
=> B = k2_funct_1(A) ) ) ) ).
fof(t64_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_funct_1(A)
& k2_relat_1(B) = k1_relat_1(A)
& k5_relat_1(B,A) = k6_relat_1(k2_relat_1(A)) )
=> B = k2_funct_1(A) ) ) ) ).
fof(t65_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
=> k2_funct_1(k2_funct_1(A)) = A ) ) ).
fof(t66_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_funct_1(A)
& v2_funct_1(B) )
=> k2_funct_1(k5_relat_1(A,B)) = k5_relat_1(k2_funct_1(B),k2_funct_1(A)) ) ) ) ).
fof(t67_funct_1,axiom,
! [A] : k2_funct_1(k6_relat_1(A)) = k6_relat_1(A) ).
fof(t68_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( B = k7_relat_1(C,A)
<=> ( k1_relat_1(B) = k3_xboole_0(k1_relat_1(C),A)
& ! [D] :
( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ).
fof(t69_funct_1,axiom,
$true ).
fof(t70_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(B,k1_relat_1(k7_relat_1(C,A)))
=> k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ).
fof(t71_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(B,k3_xboole_0(k1_relat_1(C),A))
=> k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ).
fof(t72_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(B,A)
=> k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ).
fof(t73_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(B,k1_relat_1(C))
& r2_hidden(B,A) )
=> r2_hidden(k1_funct_1(C,B),k2_relat_1(k7_relat_1(C,A))) ) ) ).
fof(t74_funct_1,axiom,
$true ).
fof(t75_funct_1,axiom,
$true ).
fof(t76_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(k1_relat_1(k7_relat_1(B,A)),k1_relat_1(B))
& r1_tarski(k2_relat_1(k7_relat_1(B,A)),k2_relat_1(B)) ) ) ).
fof(t77_funct_1,axiom,
$true ).
fof(t78_funct_1,axiom,
$true ).
fof(t79_funct_1,axiom,
$true ).
fof(t80_funct_1,axiom,
$true ).
fof(t81_funct_1,axiom,
$true ).
fof(t82_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_tarski(A,B)
=> ( k7_relat_1(k7_relat_1(C,A),B) = k7_relat_1(C,A)
& k7_relat_1(k7_relat_1(C,B),A) = k7_relat_1(C,A) ) ) ) ).
fof(t83_funct_1,axiom,
$true ).
fof(t84_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( v2_funct_1(B)
=> v2_funct_1(k7_relat_1(B,A)) ) ) ).
fof(t85_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( B = k8_relat_1(A,C)
<=> ( ! [D] :
( r2_hidden(D,k1_relat_1(B))
<=> ( r2_hidden(D,k1_relat_1(C))
& r2_hidden(k1_funct_1(C,D),A) ) )
& ! [D] :
( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ).
fof(t86_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(B,k1_relat_1(k8_relat_1(A,C)))
<=> ( r2_hidden(B,k1_relat_1(C))
& r2_hidden(k1_funct_1(C,B),A) ) ) ) ).
fof(t87_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(B,k1_relat_1(k8_relat_1(A,C)))
=> k1_funct_1(k8_relat_1(A,C),B) = k1_funct_1(C,B) ) ) ).
fof(t88_funct_1,axiom,
$true ).
fof(t89_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(k1_relat_1(k8_relat_1(A,B)),k1_relat_1(B))
& r1_tarski(k2_relat_1(k8_relat_1(A,B)),k2_relat_1(B)) ) ) ).
fof(t90_funct_1,axiom,
$true ).
fof(t91_funct_1,axiom,
$true ).
fof(t92_funct_1,axiom,
$true ).
fof(t93_funct_1,axiom,
$true ).
fof(t94_funct_1,axiom,
$true ).
fof(t95_funct_1,axiom,
$true ).
fof(t96_funct_1,axiom,
$true ).
fof(t97_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_tarski(A,B)
=> ( k8_relat_1(B,k8_relat_1(A,C)) = k8_relat_1(A,C)
& k8_relat_1(A,k8_relat_1(B,C)) = k8_relat_1(A,C) ) ) ) ).
fof(t98_funct_1,axiom,
$true ).
fof(t99_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( v2_funct_1(B)
=> v2_funct_1(k8_relat_1(A,B)) ) ) ).
fof(d10_funct_1,axiom,
$true ).
fof(d11_funct_1,axiom,
$true ).
fof(d12_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( C = k9_relat_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( r2_hidden(E,k1_relat_1(A))
& r2_hidden(E,B)
& D = k1_funct_1(A,E) ) ) ) ) ).
fof(t100_funct_1,axiom,
$true ).
fof(t101_funct_1,axiom,
$true ).
fof(t102_funct_1,axiom,
$true ).
fof(t103_funct_1,axiom,
$true ).
fof(t104_funct_1,axiom,
$true ).
fof(t105_funct_1,axiom,
$true ).
fof(t106_funct_1,axiom,
$true ).
fof(t107_funct_1,axiom,
$true ).
fof(t108_funct_1,axiom,
$true ).
fof(t109_funct_1,axiom,
$true ).
fof(t110_funct_1,axiom,
$true ).
fof(t111_funct_1,axiom,
$true ).
fof(t112_funct_1,axiom,
$true ).
fof(t113_funct_1,axiom,
$true ).
fof(t114_funct_1,axiom,
$true ).
fof(t115_funct_1,axiom,
$true ).
fof(t116_funct_1,axiom,
$true ).
fof(t117_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k1_relat_1(B))
=> k9_relat_1(B,k1_tarski(A)) = k1_tarski(k1_funct_1(B,A)) ) ) ).
fof(t118_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(A,k1_relat_1(C))
& r2_hidden(B,k1_relat_1(C)) )
=> k9_relat_1(C,k2_tarski(A,B)) = k2_tarski(k1_funct_1(C,A),k1_funct_1(C,B)) ) ) ).
fof(t119_funct_1,axiom,
$true ).
fof(t120_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> r1_tarski(k9_relat_1(k8_relat_1(A,C),B),k9_relat_1(C,B)) ) ).
fof(t121_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( v2_funct_1(C)
=> k9_relat_1(C,k3_xboole_0(A,B)) = k3_xboole_0(k9_relat_1(C,A),k9_relat_1(C,B)) ) ) ).
fof(t122_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ! [B,C] : k9_relat_1(A,k3_xboole_0(B,C)) = k3_xboole_0(k9_relat_1(A,B),k9_relat_1(A,C))
=> v2_funct_1(A) ) ) ).
fof(t123_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( v2_funct_1(C)
=> k9_relat_1(C,k4_xboole_0(A,B)) = k4_xboole_0(k9_relat_1(C,A),k9_relat_1(C,B)) ) ) ).
fof(t124_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ! [B,C] : k9_relat_1(A,k4_xboole_0(B,C)) = k4_xboole_0(k9_relat_1(A,B),k9_relat_1(A,C))
=> v2_funct_1(A) ) ) ).
fof(t125_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_xboole_0(A,B)
& v2_funct_1(C) )
=> r1_xboole_0(k9_relat_1(C,A),k9_relat_1(C,B)) ) ) ).
fof(t126_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k9_relat_1(k8_relat_1(A,C),B) = k3_xboole_0(A,k9_relat_1(C,B)) ) ).
fof(d13_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( C = k10_relat_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( r2_hidden(D,k1_relat_1(A))
& r2_hidden(k1_funct_1(A,D),B) ) ) ) ) ).
fof(t127_funct_1,axiom,
$true ).
fof(t128_funct_1,axiom,
$true ).
fof(t129_funct_1,axiom,
$true ).
fof(t130_funct_1,axiom,
$true ).
fof(t131_funct_1,axiom,
$true ).
fof(t132_funct_1,axiom,
$true ).
fof(t133_funct_1,axiom,
$true ).
fof(t134_funct_1,axiom,
$true ).
fof(t135_funct_1,axiom,
$true ).
fof(t136_funct_1,axiom,
$true ).
fof(t137_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k10_relat_1(C,k3_xboole_0(A,B)) = k3_xboole_0(k10_relat_1(C,A),k10_relat_1(C,B)) ) ).
fof(t138_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k10_relat_1(C,k4_xboole_0(A,B)) = k4_xboole_0(k10_relat_1(C,A),k10_relat_1(C,B)) ) ).
fof(t139_funct_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> k10_relat_1(k7_relat_1(C,A),B) = k3_xboole_0(A,k10_relat_1(C,B)) ) ).
fof(t140_funct_1,axiom,
$true ).
fof(t141_funct_1,axiom,
$true ).
fof(t142_funct_1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r2_hidden(A,k2_relat_1(B))
<=> k10_relat_1(B,k1_tarski(A)) != k1_xboole_0 ) ) ).
fof(t143_funct_1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( ! [C] :
~ ( r2_hidden(C,A)
& k10_relat_1(B,k1_tarski(C)) = k1_xboole_0 )
=> r1_tarski(A,k2_relat_1(B)) ) ) ).
fof(t144_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ! [B] :
~ ( r2_hidden(B,k2_relat_1(A))
& ! [C] : k10_relat_1(A,k1_tarski(B)) != k1_tarski(C) )
<=> v2_funct_1(A) ) ) ).
fof(t145_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k9_relat_1(B,k10_relat_1(B,A)),A) ) ).
fof(t146_funct_1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r1_tarski(A,k1_relat_1(B))
=> r1_tarski(A,k10_relat_1(B,k9_relat_1(B,A))) ) ) ).
fof(t147_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(A,k2_relat_1(B))
=> k9_relat_1(B,k10_relat_1(B,A)) = A ) ) ).
fof(t148_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k9_relat_1(B,k10_relat_1(B,A)) = k3_xboole_0(A,k9_relat_1(B,k1_relat_1(B))) ) ).
fof(t149_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> r1_tarski(k9_relat_1(C,k3_xboole_0(A,k10_relat_1(C,B))),k3_xboole_0(k9_relat_1(C,A),B)) ) ).
fof(t150_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k9_relat_1(C,k3_xboole_0(A,k10_relat_1(C,B))) = k3_xboole_0(k9_relat_1(C,A),B) ) ).
fof(t151_funct_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> r1_tarski(k3_xboole_0(A,k10_relat_1(C,B)),k10_relat_1(C,k3_xboole_0(k9_relat_1(C,A),B))) ) ).
fof(t152_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( v2_funct_1(B)
=> r1_tarski(k10_relat_1(B,k9_relat_1(B,A)),A) ) ) ).
fof(t153_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ! [B] : r1_tarski(k10_relat_1(A,k9_relat_1(A,B)),B)
=> v2_funct_1(A) ) ) ).
fof(t154_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( v2_funct_1(B)
=> k9_relat_1(B,A) = k10_relat_1(k2_funct_1(B),A) ) ) ).
fof(t155_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( v2_funct_1(B)
=> k10_relat_1(B,A) = k9_relat_1(k2_funct_1(B),A) ) ) ).
fof(t156_funct_1,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) )
=> ( ( A = k2_relat_1(B)
& k1_relat_1(C) = A
& k1_relat_1(D) = A
& k5_relat_1(B,C) = k5_relat_1(B,D) )
=> C = D ) ) ) ) ).
fof(t157_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(k9_relat_1(C,A),k9_relat_1(C,B))
& r1_tarski(A,k1_relat_1(C))
& v2_funct_1(C) )
=> r1_tarski(A,B) ) ) ).
fof(t158_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(k10_relat_1(C,A),k10_relat_1(C,B))
& r1_tarski(A,k2_relat_1(C)) )
=> r1_tarski(A,B) ) ) ).
fof(t159_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
<=> ! [B] :
? [C] : r1_tarski(k10_relat_1(A,k1_tarski(B)),k1_tarski(C)) ) ) ).
fof(t160_funct_1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( v1_relat_1(C)
=> ( r1_tarski(k2_relat_1(B),k1_relat_1(C))
=> r1_tarski(k10_relat_1(B,A),k10_relat_1(k5_relat_1(B,C),k9_relat_1(C,A))) ) ) ) ).
fof(t161_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( k10_relat_1(C,A) = k10_relat_1(C,B)
& r1_tarski(A,k2_relat_1(C))
& r1_tarski(B,k2_relat_1(C)) )
=> A = B ) ) ).
fof(t162_funct_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k9_relat_1(k6_relat_1(A),B) = B ) ).
fof(d14_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v3_relat_1(A)
<=> ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> v1_xboole_0(k1_funct_1(A,B)) ) ) ) ).
fof(d15_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_relat_1(A)
<=> ! [B] :
~ ( r2_hidden(B,k1_relat_1(A))
& v1_xboole_0(k1_funct_1(A,B)) ) ) ) ).
fof(s1_funct_1,axiom,
( ! [A,B,C] :
( ( p1_s1_funct_1(A,B)
& p1_s1_funct_1(A,C) )
=> B = C )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& ! [B,C] :
( r2_hidden(k4_tarski(B,C),A)
<=> ( r2_hidden(B,f1_s1_funct_1)
& p1_s1_funct_1(B,C) ) ) ) ) ).
fof(s2_funct_1,axiom,
( ( ! [A,B,C] :
( ( r2_hidden(A,f1_s2_funct_1)
& p1_s2_funct_1(A,B)
& p1_s2_funct_1(A,C) )
=> B = C )
& ! [A] :
~ ( r2_hidden(A,f1_s2_funct_1)
& ! [B] : ~ p1_s2_funct_1(A,B) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s2_funct_1
& ! [B] :
( r2_hidden(B,f1_s2_funct_1)
=> p1_s2_funct_1(B,k1_funct_1(A,B)) ) ) ) ).
fof(s3_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s3_funct_1
& ! [B] :
( r2_hidden(B,f1_s3_funct_1)
=> k1_funct_1(A,B) = f2_s3_funct_1(B) ) ) ).
fof(s4_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s4_funct_1
& ! [B] :
( m1_subset_1(B,f1_s4_funct_1)
=> k1_funct_1(A,B) = f2_s4_funct_1(B) ) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k2_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k2_funct_1(A))
& v1_funct_1(k2_funct_1(A)) ) ) ).
%------------------------------------------------------------------------------