SET007 Axioms: SET007+576.ax
%------------------------------------------------------------------------------
% File : SET007+576 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A Theory of Boolean Valued Functions and Quantifiers
% Version : [Urb08] axioms.
% English : A Theory of Boolean Valued Functions and Quantifiers with Respect
% to Partitions
% 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_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 63 ( 1 unt; 0 def)
% Number of atoms : 378 ( 43 equ)
% Maximal formula atoms : 14 ( 6 avg)
% Number of connectives : 383 ( 68 ~; 0 |; 45 &)
% ( 11 <=>; 259 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 21 ( 20 usr; 0 prp; 1-4 aty)
% Number of functors : 29 ( 29 usr; 2 con; 0-4 aty)
% Number of variables : 263 ( 257 !; 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(t1_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m1_subset_1(C,A)
=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
& r2_hidden(C,D)
& ? [E] :
( v1_relat_1(E)
& v1_funct_1(E)
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(k1_zfmisc_1(A)))
& k1_relat_1(E) = B
& k2_relat_1(E) = F
& ! [G] :
( r2_hidden(G,B)
=> r2_hidden(k1_funct_1(E,G),G) )
& D = k8_setfam_1(A,F)
& D != k1_xboole_0 ) ) ) ) ) ) ).
fof(d1_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( C = k2_bvfunc_2(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( v1_relat_1(E)
& v1_funct_1(E)
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(k1_zfmisc_1(A)))
& k1_relat_1(E) = B
& k2_relat_1(E) = F
& ! [G] :
( r2_hidden(G,B)
=> r2_hidden(k1_funct_1(E,G),G) )
& D = k8_setfam_1(A,F)
& D != k1_xboole_0 ) ) ) ) ) ) ) ).
fof(d2_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [C] :
( r1_bvfunc_2(A,B,C)
<=> ( ! [D] :
( m1_eqrel_1(D,A)
=> ( r2_hidden(D,B)
=> r1_partit1(A,D,C) ) )
& ! [D] :
( ( r1_tarski(D,C)
& ! [E] :
( m1_eqrel_1(E,A)
=> ( r2_hidden(E,B)
=> r1_partit1(A,E,D) ) ) )
=> D = C ) ) ) ) ) ).
fof(t2_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m1_subset_1(C,A)
=> ~ ( B != k1_xboole_0
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ~ ( r2_hidden(C,D)
& r1_bvfunc_2(A,B,D) ) ) ) ) ) ) ).
fof(d3_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( ( B != k1_xboole_0
=> ( C = k3_bvfunc_2(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> r1_bvfunc_2(A,B,D) ) ) )
& ( B = k1_xboole_0
=> ( C = k3_bvfunc_2(A,B)
<=> C = k3_pua2mss1(A) ) ) ) ) ) ) ).
fof(t3_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( r2_hidden(C,B)
=> r1_setfam_1(k2_bvfunc_2(A,B),C) ) ) ) ) ).
fof(t4_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( r2_hidden(C,B)
=> r1_setfam_1(C,k3_bvfunc_2(A,B)) ) ) ) ) ).
fof(d4_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ( v1_bvfunc_2(B,A)
<=> k2_bvfunc_2(A,B) = k3_pua2mss1(A) ) ) ) ).
fof(d5_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ( v2_bvfunc_2(B,A)
<=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ~ ( k1_relat_1(C) = B
& k2_relat_1(C) = D
& ! [E] :
( r2_hidden(E,B)
=> r2_hidden(k1_funct_1(C,E),E) )
& k8_setfam_1(A,D) = k1_xboole_0 ) ) ) ) ) ) ).
fof(d6_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ( r2_bvfunc_2(A,B)
<=> ( v2_bvfunc_2(B,A)
& v1_bvfunc_2(B,A) ) ) ) ) ).
fof(d7_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> k5_bvfunc_2(A,B,C) = k2_bvfunc_2(A,k6_subset_1(k1_bvfunc_2(A),C,k4_bvfunc_2(A,B))) ) ) ) ).
fof(d8_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( r3_bvfunc_2(A,B,C,D)
<=> r2_bvfunc_1(A,B,k5_bvfunc_2(A,D,C)) ) ) ) ) ) ).
fof(d9_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> k6_bvfunc_2(A,B,C,D) = k23_bvfunc_1(A,B,k5_bvfunc_2(A,D,C)) ) ) ) ) ).
fof(d10_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> k7_bvfunc_2(A,B,C,D) = k24_bvfunc_1(A,B,k5_bvfunc_2(A,D,C)) ) ) ) ) ).
fof(t5_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> r2_bvfunc_1(A,k6_bvfunc_2(A,C,B,D),k5_bvfunc_2(A,D,B)) ) ) ) ) ).
fof(t6_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> r2_bvfunc_1(A,k7_bvfunc_2(A,C,B,D),k5_bvfunc_2(A,D,B)) ) ) ) ) ).
fof(t7_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> k6_bvfunc_2(A,k19_bvfunc_1(A),B,D) = k19_bvfunc_1(A) ) ) ) ) ).
fof(t8_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> k7_bvfunc_2(A,k19_bvfunc_1(A),B,D) = k19_bvfunc_1(A) ) ) ) ) ).
fof(t9_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> k6_bvfunc_2(A,k18_bvfunc_1(A),B,D) = k18_bvfunc_1(A) ) ) ) ) ).
fof(t10_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> k7_bvfunc_2(A,k18_bvfunc_1(A),B,D) = k18_bvfunc_1(A) ) ) ) ) ).
fof(t11_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,C,B,D),C) ) ) ) ) ).
fof(t12_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> r1_bvfunc_1(A,C,k7_bvfunc_2(A,C,B,D)) ) ) ) ) ).
fof(t13_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> k6_bvfunc_2(A,k6_valuat_1(A,C,D),B,E) = k6_valuat_1(A,k6_bvfunc_2(A,C,B,E),k6_bvfunc_2(A,D,B,E)) ) ) ) ) ) ).
fof(t14_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k8_bvfunc_1(A,k6_bvfunc_2(A,C,B,E),k6_bvfunc_2(A,D,B,E)),k6_bvfunc_2(A,k8_bvfunc_1(A,C,D),B,E)) ) ) ) ) ) ).
fof(t15_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k14_bvfunc_1(A,C,D),B,E),k14_bvfunc_1(A,k6_bvfunc_2(A,C,B,E),k6_bvfunc_2(A,D,B,E))) ) ) ) ) ) ).
fof(t16_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> k7_bvfunc_2(A,k8_bvfunc_1(A,C,D),B,E) = k8_bvfunc_1(A,k7_bvfunc_2(A,C,B,E),k7_bvfunc_2(A,D,B,E)) ) ) ) ) ) ).
fof(t17_bvfunc_2,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,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k7_bvfunc_2(A,k6_valuat_1(A,B,C),D,E),k6_valuat_1(A,k7_bvfunc_2(A,B,D,E),k7_bvfunc_2(A,C,D,E))) ) ) ) ) ) ).
fof(t18_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k9_bvfunc_1(A,k7_bvfunc_2(A,C,B,E),k7_bvfunc_2(A,D,B,E)),k7_bvfunc_2(A,k9_bvfunc_1(A,C,D),B,E)) ) ) ) ) ) ).
fof(t19_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k14_bvfunc_1(A,k7_bvfunc_2(A,C,B,E),k7_bvfunc_2(A,D,B,E)),k7_bvfunc_2(A,k14_bvfunc_1(A,C,D),B,E)) ) ) ) ) ) ).
fof(t20_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> k5_valuat_1(A,k6_bvfunc_2(A,C,B,D)) = k7_bvfunc_2(A,k5_valuat_1(A,C),B,D) ) ) ) ) ).
fof(t21_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_eqrel_1(D,A)
=> k5_valuat_1(A,k7_bvfunc_2(A,C,B,D)) = k6_bvfunc_2(A,k5_valuat_1(A,C),B,D) ) ) ) ) ).
fof(t22_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> k6_bvfunc_2(A,k14_bvfunc_1(A,C,D),B,E) = k14_bvfunc_1(A,C,k6_bvfunc_2(A,D,B,E)) ) ) ) ) ) ) ).
fof(t23_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> k6_bvfunc_2(A,k14_bvfunc_1(A,D,C),B,E) = k14_bvfunc_1(A,k7_bvfunc_2(A,D,B,E),C) ) ) ) ) ) ) ).
fof(t24_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> k6_bvfunc_2(A,k8_bvfunc_1(A,C,D),B,E) = k8_bvfunc_1(A,C,k6_bvfunc_2(A,D,B,E)) ) ) ) ) ) ) ).
fof(t25_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> k6_bvfunc_2(A,k8_bvfunc_1(A,D,C),B,E) = k8_bvfunc_1(A,k6_bvfunc_2(A,D,B,E),C) ) ) ) ) ) ) ).
fof(t26_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k8_bvfunc_1(A,D,C),B,E),k8_bvfunc_1(A,k7_bvfunc_2(A,D,B,E),C)) ) ) ) ) ) ) ).
fof(t27_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> k6_bvfunc_2(A,k6_valuat_1(A,C,D),B,E) = k6_valuat_1(A,C,k6_bvfunc_2(A,D,B,E)) ) ) ) ) ) ) ).
fof(t28_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> k6_bvfunc_2(A,k6_valuat_1(A,D,C),B,E) = k6_valuat_1(A,k6_bvfunc_2(A,D,B,E),C) ) ) ) ) ) ) ).
fof(t29_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k6_valuat_1(A,C,D),B,E),k6_valuat_1(A,k7_bvfunc_2(A,C,B,E),D)) ) ) ) ) ) ).
fof(t30_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k9_bvfunc_1(A,C,D),B,E),k9_bvfunc_1(A,C,k6_bvfunc_2(A,D,B,E))) ) ) ) ) ) ) ).
fof(t31_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k9_bvfunc_1(A,D,C),B,E),k9_bvfunc_1(A,k6_bvfunc_2(A,D,B,E),C)) ) ) ) ) ) ) ).
fof(t32_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k15_bvfunc_1(A,C,D),B,E),k15_bvfunc_1(A,C,k6_bvfunc_2(A,D,B,E))) ) ) ) ) ) ) ).
fof(t33_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k15_bvfunc_1(A,D,C),B,E),k15_bvfunc_1(A,k6_bvfunc_2(A,D,B,E),C)) ) ) ) ) ) ) ).
fof(t34_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> k7_bvfunc_2(A,k8_bvfunc_1(A,C,D),B,E) = k8_bvfunc_1(A,C,k7_bvfunc_2(A,D,B,E)) ) ) ) ) ) ) ).
fof(t35_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> k7_bvfunc_2(A,k8_bvfunc_1(A,D,C),B,E) = k8_bvfunc_1(A,k7_bvfunc_2(A,D,B,E),C) ) ) ) ) ) ) ).
fof(t36_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> k7_bvfunc_2(A,k6_valuat_1(A,C,D),B,E) = k6_valuat_1(A,C,k7_bvfunc_2(A,D,B,E)) ) ) ) ) ) ) ).
fof(t37_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> k7_bvfunc_2(A,k6_valuat_1(A,D,C),B,E) = k6_valuat_1(A,k7_bvfunc_2(A,D,B,E),C) ) ) ) ) ) ) ).
fof(t38_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k14_bvfunc_1(A,C,k7_bvfunc_2(A,D,B,E)),k7_bvfunc_2(A,k14_bvfunc_1(A,C,D),B,E)) ) ) ) ) ) ).
fof(t39_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k14_bvfunc_1(A,k7_bvfunc_2(A,C,B,E),D),k7_bvfunc_2(A,k14_bvfunc_1(A,C,D),B,E)) ) ) ) ) ) ).
fof(t40_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> r1_bvfunc_1(A,k9_bvfunc_1(A,C,k7_bvfunc_2(A,D,B,E)),k7_bvfunc_2(A,k9_bvfunc_1(A,C,D),B,E)) ) ) ) ) ) ) ).
fof(t41_bvfunc_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,C,B,E)
=> r1_bvfunc_1(A,k9_bvfunc_1(A,k7_bvfunc_2(A,D,B,E),C),k7_bvfunc_2(A,k9_bvfunc_1(A,D,C),B,E)) ) ) ) ) ) ) ).
fof(dt_m1_bvfunc_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_t_1topsp(B,A)
& m1_t_1topsp(B,A) )
=> ! [C] :
( m1_bvfunc_2(C,A,B)
=> m1_eqrel_1(C,A) ) ) ).
fof(existence_m1_bvfunc_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_t_1topsp(B,A)
& m1_t_1topsp(B,A) )
=> ? [C] : m1_bvfunc_2(C,A,B) ) ).
fof(redefinition_m1_bvfunc_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_t_1topsp(B,A)
& m1_t_1topsp(B,A) )
=> ! [C] :
( m1_bvfunc_2(C,A,B)
<=> m1_subset_1(C,B) ) ) ).
fof(dt_k1_bvfunc_2,axiom,
! [A] :
( v1_t_1topsp(k1_bvfunc_2(A),A)
& m1_t_1topsp(k1_bvfunc_2(A),A) ) ).
fof(redefinition_k1_bvfunc_2,axiom,
! [A] : k1_bvfunc_2(A) = k1_partit1(A) ).
fof(dt_k2_bvfunc_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A))) )
=> m1_eqrel_1(k2_bvfunc_2(A,B),A) ) ).
fof(dt_k3_bvfunc_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A))) )
=> m1_eqrel_1(k3_bvfunc_2(A,B),A) ) ).
fof(dt_k4_bvfunc_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_eqrel_1(B,A) )
=> m1_subset_1(k4_bvfunc_2(A,B),k1_zfmisc_1(k1_bvfunc_2(A))) ) ).
fof(redefinition_k4_bvfunc_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_eqrel_1(B,A) )
=> k4_bvfunc_2(A,B) = k1_tarski(B) ) ).
fof(dt_k5_bvfunc_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_eqrel_1(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A))) )
=> m1_eqrel_1(k5_bvfunc_2(A,B,C),A) ) ).
fof(dt_k6_bvfunc_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
& m1_eqrel_1(D,A) )
=> m2_fraenkel(k6_bvfunc_2(A,B,C,D),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
fof(dt_k7_bvfunc_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_fraenkel(A,k6_margrel1))
& m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
& m1_eqrel_1(D,A) )
=> m2_fraenkel(k7_bvfunc_2(A,B,C,D),A,k6_margrel1,k1_fraenkel(A,k6_margrel1)) ) ).
%------------------------------------------------------------------------------