SET007 Axioms: SET007+576.ax


%------------------------------------------------------------------------------
% File     : SET007+576 : TPTP v7.5.0. 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 unit)
%            Number of atoms       :  378 (  43 equality)
%            Maximal formula depth :   20 (  10 average)
%            Number of connectives :  383 (  68 ~  ;   0  |;  45  &)
%                                         (  11 <=>; 259 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   0 propositional; 1-4 arity)
%            Number of functors    :   29 (   2 constant; 0-4 arity)
%            Number of variables   :  263 (   0 singleton; 257 !;   6 ?)
%            Maximal term depth    :    4 (   1 average)
% 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)) ) )).
%------------------------------------------------------------------------------