SET007 Axioms: SET007+569.ax
%------------------------------------------------------------------------------
% File : SET007+569 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A Theory of Boolean Valued Functions and Partitions
% 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 : bvfunc_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 133 ( 7 unt; 0 def)
% Number of atoms : 655 ( 129 equ)
% Maximal formula atoms : 14 ( 4 avg)
% Number of connectives : 626 ( 104 ~; 1 |; 219 &)
% ( 24 <=>; 278 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-4 aty)
% Number of functors : 58 ( 58 usr; 8 con; 0-4 aty)
% Number of variables : 360 ( 354 !; 6 ?)
% 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_bvfunc_1,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> v2_margrel1(k1_bvfunc_1(A,B)) ) ).
fof(fc2_bvfunc_1,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> v2_margrel1(k2_bvfunc_1(A,B)) ) ).
fof(cc1_bvfunc_1,axiom,
! [A] :
( v2_margrel1(A)
=> ( v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A) ) ) ).
fof(fc3_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(k3_bvfunc_1(A))
& v1_fraenkel(k3_bvfunc_1(A)) ) ).
fof(cc2_bvfunc_1,axiom,
! [A,B] :
( m1_subset_1(B,k3_bvfunc_1(A))
=> ( v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) ) ) ).
fof(fc4_bvfunc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ( v1_relat_1(k6_bvfunc_1(A,B))
& v1_funct_1(k6_bvfunc_1(A,B))
& v1_valuat_1(k6_bvfunc_1(A,B)) ) ) ).
fof(fc5_bvfunc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ( v1_relat_1(k7_bvfunc_1(A,B))
& v1_funct_1(k7_bvfunc_1(A,B))
& v1_valuat_1(k7_bvfunc_1(A,B)) ) ) ).
fof(fc6_bvfunc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ( v1_relat_1(k12_bvfunc_1(A,B))
& v1_funct_1(k12_bvfunc_1(A,B))
& v1_valuat_1(k12_bvfunc_1(A,B)) ) ) ).
fof(fc7_bvfunc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ( v1_relat_1(k13_bvfunc_1(A,B))
& v1_funct_1(k13_bvfunc_1(A,B))
& v1_valuat_1(k13_bvfunc_1(A,B)) ) ) ).
fof(fc8_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_relat_1(k18_bvfunc_1(A))
& v1_funct_1(k18_bvfunc_1(A))
& v1_funct_2(k18_bvfunc_1(A),A,k6_margrel1)
& v5_seqm_3(k18_bvfunc_1(A))
& v1_valuat_1(k18_bvfunc_1(A)) ) ) ).
fof(fc9_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_relat_1(k19_bvfunc_1(A))
& v1_funct_1(k19_bvfunc_1(A))
& v1_funct_2(k19_bvfunc_1(A),A,k6_margrel1)
& v5_seqm_3(k19_bvfunc_1(A))
& v1_valuat_1(k19_bvfunc_1(A)) ) ) ).
fof(rc1_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,A,k6_margrel1)
& v5_seqm_3(B)
& v1_valuat_1(B) ) ) ).
fof(d1_bvfunc_1,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k1_bvfunc_1(A,B) = k1_binarith(k9_margrel1(A),B) ) ) ).
fof(d2_bvfunc_1,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> k2_bvfunc_1(A,B) = k9_margrel1(k2_binarith(A,B)) ) ) ).
fof(d3_bvfunc_1,axiom,
! [A] :
( v2_margrel1(A)
=> ! [B] :
( v2_margrel1(B)
=> ( r1_xreal_0(A,B)
<=> k1_bvfunc_1(A,B) = k8_margrel1 ) ) ) ).
fof(d4_bvfunc_1,axiom,
! [A] : k3_bvfunc_1(A) = k1_fraenkel(A,k6_margrel1) ).
fof(d5_bvfunc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k6_bvfunc_1(A,B)
<=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k1_binarith(k1_funct_1(A,D),k1_funct_1(B,D)) ) ) ) ) ) ) ).
fof(d6_bvfunc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k7_bvfunc_1(A,B)
<=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k2_binarith(k1_funct_1(A,D),k1_funct_1(B,D)) ) ) ) ) ) ) ).
fof(d7_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( D = k8_bvfunc_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> k8_funct_2(A,k6_margrel1,D,E) = k3_binarith(k8_funct_2(A,k6_margrel1,B,E),k8_funct_2(A,k6_margrel1,C,E)) ) ) ) ) ) ) ).
fof(d8_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( D = k9_bvfunc_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> k8_funct_2(A,k6_margrel1,D,E) = k4_binarith(k8_funct_2(A,k6_margrel1,B,E),k8_funct_2(A,k6_margrel1,C,E)) ) ) ) ) ) ) ).
fof(d9_bvfunc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k12_bvfunc_1(A,B)
<=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k1_bvfunc_1(k1_funct_1(A,D),k1_funct_1(B,D)) ) ) ) ) ) ) ).
fof(d10_bvfunc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k13_bvfunc_1(A,B)
<=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k2_bvfunc_1(k1_funct_1(A,D),k1_funct_1(B,D)) ) ) ) ) ) ) ).
fof(d11_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( D = k14_bvfunc_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> k8_funct_2(A,k6_margrel1,D,E) = k1_binarith(k9_margrel1(k1_funct_1(B,E)),k1_funct_1(C,E)) ) ) ) ) ) ) ).
fof(d12_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( D = k15_bvfunc_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> k8_funct_2(A,k6_margrel1,D,E) = k9_margrel1(k2_binarith(k1_funct_1(B,E),k1_funct_1(C,E))) ) ) ) ) ) ) ).
fof(d13_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( B = k18_bvfunc_1(A)
<=> ! [C] :
( m1_subset_1(C,A)
=> k1_funct_1(B,C) = k7_margrel1 ) ) ) ) ).
fof(d14_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( B = k19_bvfunc_1(A)
<=> ! [C] :
( m1_subset_1(C,A)
=> k1_funct_1(B,C) = k8_margrel1 ) ) ) ) ).
fof(t1_bvfunc_1,axiom,
$true ).
fof(t2_bvfunc_1,axiom,
$true ).
fof(t3_bvfunc_1,axiom,
$true ).
fof(t4_bvfunc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A) )
=> k3_valuat_1(k3_valuat_1(A)) = A ) ).
fof(t5_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( k5_valuat_1(A,k19_bvfunc_1(A)) = k18_bvfunc_1(A)
& k5_valuat_1(A,k18_bvfunc_1(A)) = k19_bvfunc_1(A) ) ) ) ).
fof(t6_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k6_valuat_1(A,B,B) = B ) ) ) ).
fof(t7_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k6_valuat_1(A,k6_valuat_1(A,B,C),D) = k6_valuat_1(A,B,k6_valuat_1(A,C,D)) ) ) ) ) ).
fof(t8_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k6_valuat_1(A,B,k18_bvfunc_1(A)) = k18_bvfunc_1(A) ) ) ).
fof(t9_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k6_valuat_1(A,B,k19_bvfunc_1(A)) = B ) ) ).
fof(t10_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k8_bvfunc_1(A,B,B) = B ) ) ).
fof(t11_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k8_bvfunc_1(A,k8_bvfunc_1(A,B,C),D) = k8_bvfunc_1(A,B,k8_bvfunc_1(A,C,D)) ) ) ) ) ).
fof(t12_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k8_bvfunc_1(A,B,k18_bvfunc_1(A)) = B ) ) ).
fof(t13_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k8_bvfunc_1(A,B,k19_bvfunc_1(A)) = k19_bvfunc_1(A) ) ) ).
fof(t14_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k8_bvfunc_1(A,k6_valuat_1(A,B,C),D) = k6_valuat_1(A,k8_bvfunc_1(A,B,D),k8_bvfunc_1(A,C,D)) ) ) ) ) ).
fof(t15_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k6_valuat_1(A,k8_bvfunc_1(A,B,C),D) = k8_bvfunc_1(A,k6_valuat_1(A,B,D),k6_valuat_1(A,C,D)) ) ) ) ) ).
fof(t16_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k5_valuat_1(A,k8_bvfunc_1(A,B,C)) = k6_valuat_1(A,k5_valuat_1(A,B),k5_valuat_1(A,C)) ) ) ) ).
fof(t17_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k5_valuat_1(A,k6_valuat_1(A,B,C)) = k8_bvfunc_1(A,k5_valuat_1(A,B),k5_valuat_1(A,C)) ) ) ) ).
fof(d15_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( r1_bvfunc_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> ( k1_funct_1(B,D) = k8_margrel1
=> k1_funct_1(C,D) = k8_margrel1 ) ) ) ) ) ) ).
fof(t18_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( ( ( r1_bvfunc_1(A,B,C)
& r1_bvfunc_1(A,C,B) )
=> B = C )
& ( ( r1_bvfunc_1(A,B,C)
& r1_bvfunc_1(A,C,D) )
=> r1_bvfunc_1(A,B,D) ) ) ) ) ) ) ).
fof(t19_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( k14_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
<=> r1_bvfunc_1(A,B,C) ) ) ) ) ).
fof(t20_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
<=> B = C ) ) ) ) ).
fof(t21_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( r1_bvfunc_1(A,k18_bvfunc_1(A),B)
& r1_bvfunc_1(A,B,k19_bvfunc_1(A)) ) ) ) ).
fof(d16_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( ( ! [D] :
( m1_subset_1(D,A)
=> k1_funct_1(B,D) = k8_margrel1 )
=> ( C = k20_bvfunc_1(A,B)
<=> C = k19_bvfunc_1(A) ) )
& ( ~ ! [D] :
( m1_subset_1(D,A)
=> k1_funct_1(B,D) = k8_margrel1 )
=> ( C = k20_bvfunc_1(A,B)
<=> C = k18_bvfunc_1(A) ) ) ) ) ) ) ).
fof(d17_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( ( ! [D] :
( m1_subset_1(D,A)
=> k1_funct_1(B,D) = k7_margrel1 )
=> ( C = k21_bvfunc_1(A,B)
<=> C = k18_bvfunc_1(A) ) )
& ( ~ ! [D] :
( m1_subset_1(D,A)
=> k1_funct_1(B,D) = k7_margrel1 )
=> ( C = k21_bvfunc_1(A,B)
<=> C = k19_bvfunc_1(A) ) ) ) ) ) ) ).
fof(t22_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( k5_valuat_1(A,k20_bvfunc_1(A,B)) = k21_bvfunc_1(A,k5_valuat_1(A,B))
& k5_valuat_1(A,k21_bvfunc_1(A,B)) = k20_bvfunc_1(A,k5_valuat_1(A,B)) ) ) ) ).
fof(t23_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( k20_bvfunc_1(A,k18_bvfunc_1(A)) = k18_bvfunc_1(A)
& k20_bvfunc_1(A,k19_bvfunc_1(A)) = k19_bvfunc_1(A)
& k21_bvfunc_1(A,k18_bvfunc_1(A)) = k18_bvfunc_1(A)
& k21_bvfunc_1(A,k19_bvfunc_1(A)) = k19_bvfunc_1(A) ) ) ).
fof(t24_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v5_seqm_3(B)
& m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) )
=> ( B = k18_bvfunc_1(A)
| B = k19_bvfunc_1(A) ) ) ) ).
fof(t25_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v5_seqm_3(B)
& m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) )
=> ( k20_bvfunc_1(A,B) = B
& k21_bvfunc_1(A,B) = B ) ) ) ).
fof(t26_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( k20_bvfunc_1(A,k6_valuat_1(A,B,C)) = k6_valuat_1(A,k20_bvfunc_1(A,B),k20_bvfunc_1(A,C))
& k21_bvfunc_1(A,k8_bvfunc_1(A,B,C)) = k8_bvfunc_1(A,k21_bvfunc_1(A,B),k21_bvfunc_1(A,C)) ) ) ) ) ).
fof(t27_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( ( v5_seqm_3(C)
& m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) )
=> ( k20_bvfunc_1(A,k14_bvfunc_1(A,C,B)) = k14_bvfunc_1(A,C,k20_bvfunc_1(A,B))
& k20_bvfunc_1(A,k14_bvfunc_1(A,B,C)) = k14_bvfunc_1(A,k21_bvfunc_1(A,B),C) ) ) ) ) ).
fof(t28_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( ( v5_seqm_3(C)
& m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) )
=> ( k20_bvfunc_1(A,k8_bvfunc_1(A,C,B)) = k8_bvfunc_1(A,C,k20_bvfunc_1(A,B))
& k21_bvfunc_1(A,k6_valuat_1(A,C,B)) = k6_valuat_1(A,C,k21_bvfunc_1(A,B))
& k21_bvfunc_1(A,k6_valuat_1(A,B,C)) = k6_valuat_1(A,k21_bvfunc_1(A,B),C) ) ) ) ) ).
fof(t29_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,A)
=> r1_xreal_0(k1_funct_1(k20_bvfunc_1(A,B),C),k1_funct_1(B,C)) ) ) ) ).
fof(t30_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,A)
=> r1_xreal_0(k1_funct_1(B,C),k1_funct_1(k21_bvfunc_1(A,B),C)) ) ) ) ).
fof(d18_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( r2_bvfunc_1(A,B,C)
<=> ! [D] :
( r2_hidden(D,C)
=> ! [E,F] :
( ( r2_hidden(E,D)
& r2_hidden(F,D) )
=> k1_funct_1(B,E) = k1_funct_1(B,F) ) ) ) ) ) ) ).
fof(t31_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> r2_bvfunc_1(A,B,k3_pua2mss1(A)) ) ) ).
fof(t32_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v5_seqm_3(B)
& m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) )
=> r2_bvfunc_1(A,B,k6_partit1(A)) ) ) ).
fof(d19_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( D = k23_bvfunc_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> ( ( ! [F] :
( m1_subset_1(F,A)
=> ( r2_hidden(F,k22_bvfunc_1(A,E,C))
=> k1_funct_1(B,F) = k8_margrel1 ) )
=> k1_funct_1(D,E) = k8_margrel1 )
& ( ? [F] :
( m1_subset_1(F,A)
& r2_hidden(F,k22_bvfunc_1(A,E,C))
& k1_funct_1(B,F) != k8_margrel1 )
=> k1_funct_1(D,E) = k7_margrel1 ) ) ) ) ) ) ) ) ).
fof(d20_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( D = k24_bvfunc_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> ( ( ? [F] :
( m1_subset_1(F,A)
& r2_hidden(F,k22_bvfunc_1(A,E,C))
& k1_funct_1(B,F) = k8_margrel1 )
=> k1_funct_1(D,E) = k8_margrel1 )
& ( ! [F] :
( m1_subset_1(F,A)
=> ~ ( r2_hidden(F,k22_bvfunc_1(A,E,C))
& k1_funct_1(B,F) = k8_margrel1 ) )
=> k1_funct_1(D,E) = k7_margrel1 ) ) ) ) ) ) ) ) ).
fof(t33_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_eqrel_1(C,A)
=> r2_bvfunc_1(A,k23_bvfunc_1(A,B,C),C) ) ) ) ).
fof(t34_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_eqrel_1(C,A)
=> r2_bvfunc_1(A,k24_bvfunc_1(A,B,C),C) ) ) ) ).
fof(t35_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_eqrel_1(C,A)
=> r1_bvfunc_1(A,k23_bvfunc_1(A,B,C),B) ) ) ) ).
fof(t36_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_eqrel_1(C,A)
=> r1_bvfunc_1(A,B,k24_bvfunc_1(A,B,C)) ) ) ) ).
fof(t37_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_eqrel_1(C,A)
=> k5_valuat_1(A,k23_bvfunc_1(A,B,C)) = k24_bvfunc_1(A,k5_valuat_1(A,B),C) ) ) ) ).
fof(t38_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k23_bvfunc_1(A,B,k6_partit1(A)) = k20_bvfunc_1(A,B) ) ) ).
fof(t39_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k24_bvfunc_1(A,B,k6_partit1(A)) = k21_bvfunc_1(A,B) ) ) ).
fof(t40_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k23_bvfunc_1(A,B,k3_pua2mss1(A)) = B ) ) ).
fof(t41_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k24_bvfunc_1(A,B,k3_pua2mss1(A)) = B ) ) ).
fof(t42_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> k23_bvfunc_1(A,k6_valuat_1(A,B,C),D) = k6_valuat_1(A,k23_bvfunc_1(A,B,D),k23_bvfunc_1(A,C,D)) ) ) ) ) ).
fof(t43_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> k24_bvfunc_1(A,k8_bvfunc_1(A,B,C),D) = k8_bvfunc_1(A,k24_bvfunc_1(A,B,D),k24_bvfunc_1(A,C,D)) ) ) ) ) ).
fof(t44_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> r2_bvfunc_1(A,B,k25_bvfunc_1(A,B)) ) ) ).
fof(t45_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( r2_bvfunc_1(A,B,C)
=> r1_setfam_1(C,k25_bvfunc_1(A,B)) ) ) ) ) ).
fof(s1_bvfunc_1,axiom,
! [A] :
( m2_fraenkel(A,f1_s1_bvfunc_1,k6_margrel1,k1_fraenkel(f1_s1_bvfunc_1,k6_margrel1))
=> ! [B] :
( m2_fraenkel(B,f1_s1_bvfunc_1,k6_margrel1,k1_fraenkel(f1_s1_bvfunc_1,k6_margrel1))
=> ( ( ! [C] :
( m1_subset_1(C,f1_s1_bvfunc_1)
=> k1_funct_1(A,C) = f4_s1_bvfunc_1(C,f2_s1_bvfunc_1,f3_s1_bvfunc_1) )
& ! [C] :
( m1_subset_1(C,f1_s1_bvfunc_1)
=> k1_funct_1(B,C) = f4_s1_bvfunc_1(C,f2_s1_bvfunc_1,f3_s1_bvfunc_1) ) )
=> A = B ) ) ) ).
fof(s2_bvfunc_1,axiom,
! [A] :
( m2_fraenkel(A,f1_s2_bvfunc_1,k6_margrel1,k1_fraenkel(f1_s2_bvfunc_1,k6_margrel1))
=> ! [B] :
( m2_fraenkel(B,f1_s2_bvfunc_1,k6_margrel1,k1_fraenkel(f1_s2_bvfunc_1,k6_margrel1))
=> ( ( ! [C] :
( m1_subset_1(C,f1_s2_bvfunc_1)
=> k1_funct_1(A,C) = f2_s2_bvfunc_1(C) )
& ! [C] :
( m1_subset_1(C,f1_s2_bvfunc_1)
=> k1_funct_1(B,C) = f2_s2_bvfunc_1(C) ) )
=> A = B ) ) ) ).
fof(dt_m1_bvfunc_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_eqrel_1(B,A) )
=> ! [C] :
( m1_bvfunc_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(A)) ) ) ).
fof(existence_m1_bvfunc_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_eqrel_1(B,A) )
=> ? [C] : m1_bvfunc_1(C,A,B) ) ).
fof(redefinition_m1_bvfunc_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_eqrel_1(B,A) )
=> ! [C] :
( m1_bvfunc_1(C,A,B)
<=> m1_subset_1(C,B) ) ) ).
fof(reflexivity_r1_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> r1_bvfunc_1(A,B,B) ) ).
fof(dt_k1_bvfunc_1,axiom,
$true ).
fof(dt_k2_bvfunc_1,axiom,
$true ).
fof(commutativity_k2_bvfunc_1,axiom,
! [A,B] :
( ( v2_margrel1(A)
& v2_margrel1(B) )
=> k2_bvfunc_1(A,B) = k2_bvfunc_1(B,A) ) ).
fof(dt_k3_bvfunc_1,axiom,
$true ).
fof(dt_k4_bvfunc_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A)) )
=> m1_subset_1(k4_bvfunc_1(A,B),k3_bvfunc_1(A)) ) ).
fof(redefinition_k4_bvfunc_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A)) )
=> k4_bvfunc_1(A,B) = k3_valuat_1(B) ) ).
fof(dt_k5_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> m1_subset_1(k5_bvfunc_1(A,B,C),k3_bvfunc_1(A)) ) ).
fof(commutativity_k5_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> k5_bvfunc_1(A,B,C) = k5_bvfunc_1(A,C,B) ) ).
fof(redefinition_k5_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> k5_bvfunc_1(A,B,C) = k4_valuat_1(B,C) ) ).
fof(dt_k6_bvfunc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ( v1_relat_1(k6_bvfunc_1(A,B))
& v1_funct_1(k6_bvfunc_1(A,B)) ) ) ).
fof(commutativity_k6_bvfunc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> k6_bvfunc_1(A,B) = k6_bvfunc_1(B,A) ) ).
fof(dt_k7_bvfunc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ( v1_relat_1(k7_bvfunc_1(A,B))
& v1_funct_1(k7_bvfunc_1(A,B)) ) ) ).
fof(commutativity_k7_bvfunc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> k7_bvfunc_1(A,B) = k7_bvfunc_1(B,A) ) ).
fof(dt_k8_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> m2_fraenkel(k8_bvfunc_1(A,B,C),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
fof(commutativity_k8_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> k8_bvfunc_1(A,B,C) = k8_bvfunc_1(A,C,B) ) ).
fof(redefinition_k8_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> k8_bvfunc_1(A,B,C) = k6_bvfunc_1(B,C) ) ).
fof(dt_k9_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> m2_fraenkel(k9_bvfunc_1(A,B,C),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
fof(commutativity_k9_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> k9_bvfunc_1(A,B,C) = k9_bvfunc_1(A,C,B) ) ).
fof(redefinition_k9_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> k9_bvfunc_1(A,B,C) = k7_bvfunc_1(B,C) ) ).
fof(dt_k10_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> m1_subset_1(k10_bvfunc_1(A,B,C),k3_bvfunc_1(A)) ) ).
fof(commutativity_k10_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> k10_bvfunc_1(A,B,C) = k10_bvfunc_1(A,C,B) ) ).
fof(redefinition_k10_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> k10_bvfunc_1(A,B,C) = k6_bvfunc_1(B,C) ) ).
fof(dt_k11_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> m1_subset_1(k11_bvfunc_1(A,B,C),k3_bvfunc_1(A)) ) ).
fof(commutativity_k11_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> k11_bvfunc_1(A,B,C) = k11_bvfunc_1(A,C,B) ) ).
fof(redefinition_k11_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> k11_bvfunc_1(A,B,C) = k7_bvfunc_1(B,C) ) ).
fof(dt_k12_bvfunc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ( v1_relat_1(k12_bvfunc_1(A,B))
& v1_funct_1(k12_bvfunc_1(A,B)) ) ) ).
fof(dt_k13_bvfunc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> ( v1_relat_1(k13_bvfunc_1(A,B))
& v1_funct_1(k13_bvfunc_1(A,B)) ) ) ).
fof(commutativity_k13_bvfunc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_valuat_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_valuat_1(B) )
=> k13_bvfunc_1(A,B) = k13_bvfunc_1(B,A) ) ).
fof(dt_k14_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> m2_fraenkel(k14_bvfunc_1(A,B,C),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
fof(redefinition_k14_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> k14_bvfunc_1(A,B,C) = k12_bvfunc_1(B,C) ) ).
fof(dt_k15_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> m2_fraenkel(k15_bvfunc_1(A,B,C),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
fof(commutativity_k15_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> k15_bvfunc_1(A,B,C) = k15_bvfunc_1(A,C,B) ) ).
fof(redefinition_k15_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_fraenkel(A,k6_margrel1)) )
=> k15_bvfunc_1(A,B,C) = k13_bvfunc_1(B,C) ) ).
fof(dt_k16_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> m1_subset_1(k16_bvfunc_1(A,B,C),k3_bvfunc_1(A)) ) ).
fof(redefinition_k16_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> k16_bvfunc_1(A,B,C) = k12_bvfunc_1(B,C) ) ).
fof(dt_k17_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> m1_subset_1(k17_bvfunc_1(A,B,C),k3_bvfunc_1(A)) ) ).
fof(commutativity_k17_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> k17_bvfunc_1(A,B,C) = k17_bvfunc_1(A,C,B) ) ).
fof(redefinition_k17_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k3_bvfunc_1(A))
& m1_subset_1(C,k3_bvfunc_1(A)) )
=> k17_bvfunc_1(A,B,C) = k13_bvfunc_1(B,C) ) ).
fof(dt_k18_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m2_fraenkel(k18_bvfunc_1(A),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
fof(dt_k19_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m2_fraenkel(k19_bvfunc_1(A),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
fof(dt_k20_bvfunc_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1)) )
=> m2_fraenkel(k20_bvfunc_1(A,B),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
fof(dt_k21_bvfunc_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1)) )
=> m2_fraenkel(k21_bvfunc_1(A,B),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
fof(dt_k22_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_eqrel_1(C,A) )
=> m1_bvfunc_1(k22_bvfunc_1(A,B,C),A,C) ) ).
fof(redefinition_k22_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_eqrel_1(C,A) )
=> k22_bvfunc_1(A,B,C) = k1_t_1topsp(A,B,C) ) ).
fof(dt_k23_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_eqrel_1(C,A) )
=> m2_fraenkel(k23_bvfunc_1(A,B,C),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
fof(dt_k24_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_eqrel_1(C,A) )
=> m2_fraenkel(k24_bvfunc_1(A,B,C),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
fof(dt_k25_bvfunc_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1)) )
=> m1_eqrel_1(k25_bvfunc_1(A,B),A) ) ).
fof(d21_bvfunc_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k25_bvfunc_1(A,B) = k4_xboole_0(k2_tarski(a_2_0_bvfunc_1(A,B),a_2_1_bvfunc_1(A,B)),k1_tarski(k1_xboole_0)) ) ) ).
fof(fraenkel_a_2_0_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m2_fraenkel(C,B,k6_margrel1,k1_fraenkel(B,k6_margrel1)) )
=> ( r2_hidden(A,a_2_0_bvfunc_1(B,C))
<=> ? [D] :
( m1_subset_1(D,B)
& A = D
& k8_funct_2(B,k6_margrel1,C,D) = k8_margrel1 ) ) ) ).
fof(fraenkel_a_2_1_bvfunc_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m2_fraenkel(C,B,k6_margrel1,k1_fraenkel(B,k6_margrel1)) )
=> ( r2_hidden(A,a_2_1_bvfunc_1(B,C))
<=> ? [D] :
( m1_subset_1(D,B)
& A = D
& k8_funct_2(B,k6_margrel1,C,D) = k7_margrel1 ) ) ) ).
%------------------------------------------------------------------------------