SET007 Axioms: SET007+580.ax


%------------------------------------------------------------------------------
% File     : SET007+580 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Lattice of Substitutions is a Heyting Algebra
% 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    : heyting2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   56 (   2 unit)
%            Number of atoms       :  290 (  49 equality)
%            Maximal formula depth :   18 (   9 average)
%            Number of connectives :  257 (  23 ~  ;   0  |;  90  &)
%                                         (  10 <=>; 134 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   1 propositional; 0-3 arity)
%            Number of functors    :   44 (   1 constant; 0-6 arity)
%            Number of variables   :  216 (   0 singleton; 210 !;   6 ?)
%            Maximal term depth    :    6 (   2 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(fc1_heyting2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(k1_tarski(k4_tarski(A,B)))
      & v1_relat_1(k1_tarski(k4_tarski(A,B)))
      & v1_funct_1(k1_tarski(k4_tarski(A,B))) ) )).

fof(rc1_heyting2,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A)
      & v1_fraenkel(A) ) )).

fof(fc2_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ( ~ v3_struct_0(k5_substlat(A,B))
        & v3_lattices(k5_substlat(A,B))
        & v4_lattices(k5_substlat(A,B))
        & v5_lattices(k5_substlat(A,B))
        & v6_lattices(k5_substlat(A,B))
        & v7_lattices(k5_substlat(A,B))
        & v8_lattices(k5_substlat(A,B))
        & v9_lattices(k5_substlat(A,B))
        & v10_lattices(k5_substlat(A,B))
        & v11_lattices(k5_substlat(A,B))
        & v12_lattices(k5_substlat(A,B))
        & v13_lattices(k5_substlat(A,B))
        & v14_lattices(k5_substlat(A,B))
        & v15_lattices(k5_substlat(A,B))
        & v3_filter_0(k5_substlat(A,B)) ) ) )).

fof(t1_heyting2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ~ ! [C] :
                ( m1_subset_1(C,k4_partfun1(A,B))
               => C = k1_xboole_0 ) ) ) )).

fof(t2_heyting2,axiom,(
    ! [A,B,C,D] :
      ( ( r2_hidden(D,k1_substlat(A,B))
        & r2_hidden(C,D) )
     => ( v1_relat_1(C)
        & v1_funct_1(C)
        & v1_finset_1(C) ) ) )).

fof(t3_heyting2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k4_partfun1(A,B))
     => ! [D] :
          ( r1_tarski(D,C)
         => r2_hidden(D,k4_partfun1(A,B)) ) ) )).

fof(t4_heyting2,axiom,(
    ! [A,B] : r1_tarski(k4_partfun1(A,B),k1_zfmisc_1(k2_zfmisc_1(A,B))) )).

fof(t5_heyting2,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k4_partfun1(A,B)) ) )).

fof(t6_heyting2,axiom,(
    ! [A,B,C] :
      ( ( v1_finset_1(C)
        & m1_subset_1(C,k4_partfun1(A,B)) )
     => r2_hidden(k2_setwiseo(k4_partfun1(A,B),C),k1_substlat(A,B)) ) )).

fof(t7_heyting2,axiom,(
    ! [A,B,C] :
      ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
     => ! [D] :
          ( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
         => ( k4_substlat(A,B,C,D) = C
           => ! [E] :
                ~ ( r2_hidden(E,C)
                  & ! [F] :
                      ~ ( r2_hidden(F,D)
                        & r1_tarski(F,E) ) ) ) ) ) )).

fof(t8_heyting2,axiom,(
    ! [A,B,C] :
      ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
     => ! [D] :
          ( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
         => ( k3_substlat(A,B,k4_substlat(A,B,C,D)) = C
           => ! [E] :
                ~ ( r2_hidden(E,C)
                  & ! [F] :
                      ~ ( r2_hidden(F,D)
                        & r1_tarski(F,E) ) ) ) ) ) )).

fof(t9_heyting2,axiom,(
    ! [A,B,C] :
      ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
     => ! [D] :
          ( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
         => ( ! [E] :
                ~ ( r2_hidden(E,C)
                  & ! [F] :
                      ~ ( r2_hidden(F,D)
                        & r1_tarski(F,E) ) )
           => k3_substlat(A,B,k4_substlat(A,B,C,D)) = C ) ) ) )).

fof(d1_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
         => ! [D] :
              ( D = k1_heyting2(A,B,C)
            <=> ! [E] :
                  ( r2_hidden(E,D)
                <=> ? [F] :
                      ( v1_relat_1(F)
                      & v1_funct_1(F)
                      & v1_finset_1(F)
                      & r2_hidden(F,C)
                      & r2_hidden(E,k1_relat_1(F)) ) ) ) ) ) )).

fof(t10_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
         => r1_tarski(k1_heyting2(A,B,C),A) ) ) )).

fof(t11_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
         => ( C = k1_xboole_0
           => k1_heyting2(A,B,C) = k1_xboole_0 ) ) ) )).

fof(t12_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
         => v1_finset_1(k1_heyting2(A,B,C)) ) ) )).

fof(t13_heyting2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m1_subset_1(B,k5_finsub_1(k4_partfun1(k1_xboole_0,A)))
         => k1_heyting2(k1_xboole_0,A,B) = k1_xboole_0 ) ) )).

fof(t14_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
         => k4_substlat(A,B,C,k2_heyting2(A,B,C)) = k1_xboole_0 ) ) )).

fof(t15_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
         => ( C = k1_xboole_0
           => k2_heyting2(A,B,C) = k1_tarski(k1_xboole_0) ) ) ) )).

fof(t16_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
         => ( C = k1_tarski(k1_xboole_0)
           => k2_heyting2(A,B,C) = k1_xboole_0 ) ) ) )).

fof(t17_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
         => k3_substlat(A,B,k4_substlat(A,B,C,k2_heyting2(A,B,C))) = k5_lattices(k5_substlat(A,B)) ) ) )).

fof(t18_heyting2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B) )
         => ! [C] :
              ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
             => ( C = k1_xboole_0
               => k3_substlat(A,B,k2_heyting2(A,B,C)) = k6_lattices(k5_substlat(A,B)) ) ) ) ) )).

fof(t19_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
         => ! [D] :
              ( m1_subset_1(D,k4_partfun1(A,B))
             => ! [E] :
                  ( m2_subset_1(E,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
                 => ~ ( E = k2_setwiseo(k4_partfun1(A,B),D)
                      & k4_substlat(A,B,C,E) = k1_xboole_0
                      & ! [F] :
                          ( v1_finset_1(F)
                         => ~ ( r2_hidden(F,k2_heyting2(A,B,C))
                              & r1_tarski(F,D) ) ) ) ) ) ) ) )).

fof(t21_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
         => ( C = k1_xboole_0
           => k3_heyting2(A,B,C,C) = k1_tarski(k1_xboole_0) ) ) ) )).

fof(t22_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
         => ! [D] :
              ( r1_tarski(D,C)
             => m1_subset_1(D,u1_struct_0(k5_substlat(A,B))) ) ) ) )).

fof(d4_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)))
            & m2_relset_1(C,u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))) )
         => ( C = k4_heyting2(A,B)
          <=> ! [D] :
                ( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
               => ! [E] :
                    ( m2_subset_1(E,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
                   => ( E = D
                     => k8_funct_2(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),C,D) = k3_substlat(A,B,k2_heyting2(A,B,E)) ) ) ) ) ) ) )).

fof(d5_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))),u1_struct_0(k5_substlat(A,B)))
            & m2_relset_1(C,k2_zfmisc_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))),u1_struct_0(k5_substlat(A,B))) )
         => ( C = k5_heyting2(A,B)
          <=> ! [D] :
                ( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
               => ! [E] :
                    ( m1_subset_1(E,u1_struct_0(k5_substlat(A,B)))
                   => ! [F] :
                        ( m2_subset_1(F,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
                       => ! [G] :
                            ( m2_subset_1(G,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
                           => ( ( F = D
                                & G = E )
                             => k2_binop_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),C,D,E) = k3_substlat(A,B,k3_heyting2(A,B,F,G)) ) ) ) ) ) ) ) ) )).

fof(d6_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
         => k6_heyting2(A,B,C) = k1_zfmisc_1(C) ) ) )).

fof(d7_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)))
                & m2_relset_1(D,u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))) )
             => ( D = k7_heyting2(A,B,C)
              <=> ! [E] :
                    ( m1_subset_1(E,u1_struct_0(k5_substlat(A,B)))
                   => k8_funct_2(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),D,E) = k4_xboole_0(C,E) ) ) ) ) ) )).

fof(d8_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)))
            & m2_relset_1(C,k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B))) )
         => ( C = k8_heyting2(A,B)
          <=> ! [D] :
                ( m1_subset_1(D,k4_partfun1(A,B))
               => k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),C,D) = k3_substlat(A,B,k2_setwiseo(k4_partfun1(A,B),D)) ) ) ) ) )).

fof(t23_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
         => k2_lattice2(k4_partfun1(A,B),k5_substlat(A,B),C,k8_heyting2(A,B)) = k10_setwiseo(k4_partfun1(A,B),k4_partfun1(A,B),C,k11_setwiseo(k4_partfun1(A,B))) ) ) )).

fof(t24_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
         => C = k2_lattice2(k4_partfun1(A,B),k5_substlat(A,B),C,k8_heyting2(A,B)) ) ) )).

fof(t25_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
             => r3_lattices(k5_substlat(A,B),k8_funct_2(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k7_heyting2(A,B,C),D),C) ) ) ) )).

fof(t26_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,k4_partfun1(A,B))
         => ( v1_finset_1(C)
           => ! [D] :
                ( r2_hidden(D,k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),k8_heyting2(A,B),C))
               => D = C ) ) ) ) )).

fof(t27_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
         => ! [D] :
              ( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
             => ! [E] :
                  ( m2_subset_1(E,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
                 => ! [F] :
                      ( m1_subset_1(F,k4_partfun1(A,B))
                     => ( ( D = k2_setwiseo(k4_partfun1(A,B),F)
                          & E = C
                          & k4_substlat(A,B,E,D) = k1_xboole_0 )
                       => r3_lattices(k5_substlat(A,B),k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),k8_heyting2(A,B),F),k8_funct_2(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k4_heyting2(A,B),C)) ) ) ) ) ) ) )).

fof(t28_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( ( v1_finset_1(C)
            & m1_subset_1(C,k4_partfun1(A,B)) )
         => r2_hidden(C,k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),k8_heyting2(A,B),C)) ) ) )).

fof(t29_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,k4_partfun1(A,B))
         => ! [D] :
              ( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
             => ! [E] :
                  ( m2_subset_1(E,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
                 => ~ ( ! [F] :
                          ~ ( r2_hidden(F,D)
                            & ! [G] :
                                ~ ( r2_hidden(G,E)
                                  & r1_tarski(G,k2_xboole_0(F,C)) ) )
                      & ! [F] :
                          ~ ( r2_hidden(F,k3_heyting2(A,B,D,E))
                            & r1_tarski(F,C) ) ) ) ) ) ) )).

fof(t30_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
             => ! [E] :
                  ( ( v1_finset_1(E)
                    & m1_subset_1(E,k4_partfun1(A,B)) )
                 => ( ( ! [F] :
                          ( m1_subset_1(F,k4_partfun1(A,B))
                         => ( r2_hidden(F,C)
                           => r1_partfun1(F,E) ) )
                      & r3_lattices(k5_substlat(A,B),k4_lattices(k5_substlat(A,B),C,k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),k8_heyting2(A,B),E)),D) )
                   => r3_lattices(k5_substlat(A,B),k8_funct_2(k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)),k8_heyting2(A,B),E),k2_binop_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k5_heyting2(A,B),C,D)) ) ) ) ) ) )).

fof(t31_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
         => k4_lattices(k5_substlat(A,B),C,k8_funct_2(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k4_heyting2(A,B),C)) = k5_lattices(k5_substlat(A,B)) ) ) )).

fof(t32_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
             => r3_lattices(k5_substlat(A,B),k4_lattices(k5_substlat(A,B),C,k2_binop_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k5_heyting2(A,B),C,D)),D) ) ) ) )).

fof(t33_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(k5_substlat(A,B)))
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(k5_substlat(A,B)))
             => k4_filter_0(k5_substlat(A,B),C,D) = k2_lattice2(u1_struct_0(k5_substlat(A,B)),k5_substlat(A,B),k6_heyting2(A,B,C),k6_funcop_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),u1_lattices(k5_substlat(A,B)),k4_heyting2(A,B),k7_funcop_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)),k5_heyting2(A,B),k7_heyting2(A,B,C),D))) ) ) ) )).

fof(dt_k1_heyting2,axiom,(
    $true )).

fof(dt_k2_heyting2,axiom,(
    ! [A,B,C] :
      ( ( v1_finset_1(B)
        & m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B))) )
     => m1_subset_1(k2_heyting2(A,B,C),k5_finsub_1(k4_partfun1(A,B))) ) )).

fof(dt_k3_heyting2,axiom,(
    ! [A,B,C,D] :
      ( ( v1_finset_1(B)
        & m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
        & m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B))) )
     => m1_subset_1(k3_heyting2(A,B,C,D),k5_finsub_1(k4_partfun1(A,B))) ) )).

fof(dt_k4_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ( v1_funct_1(k4_heyting2(A,B))
        & v1_funct_2(k4_heyting2(A,B),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)))
        & m2_relset_1(k4_heyting2(A,B),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))) ) ) )).

fof(dt_k5_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ( v1_funct_1(k5_heyting2(A,B))
        & v1_funct_2(k5_heyting2(A,B),k2_zfmisc_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))),u1_struct_0(k5_substlat(A,B)))
        & m2_relset_1(k5_heyting2(A,B),k2_zfmisc_1(u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))),u1_struct_0(k5_substlat(A,B))) ) ) )).

fof(dt_k6_heyting2,axiom,(
    ! [A,B,C] :
      ( ( v1_finset_1(B)
        & m1_subset_1(C,u1_struct_0(k5_substlat(A,B))) )
     => m1_subset_1(k6_heyting2(A,B,C),k5_finsub_1(u1_struct_0(k5_substlat(A,B)))) ) )).

fof(dt_k7_heyting2,axiom,(
    ! [A,B,C] :
      ( ( v1_finset_1(B)
        & m1_subset_1(C,u1_struct_0(k5_substlat(A,B))) )
     => ( v1_funct_1(k7_heyting2(A,B,C))
        & v1_funct_2(k7_heyting2(A,B,C),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B)))
        & m2_relset_1(k7_heyting2(A,B,C),u1_struct_0(k5_substlat(A,B)),u1_struct_0(k5_substlat(A,B))) ) ) )).

fof(dt_k8_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ( v1_funct_1(k8_heyting2(A,B))
        & v1_funct_2(k8_heyting2(A,B),k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B)))
        & m2_relset_1(k8_heyting2(A,B),k4_partfun1(A,B),u1_struct_0(k5_substlat(A,B))) ) ) )).

fof(d2_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
         => k2_heyting2(A,B,C) = a_3_0_heyting2(A,B,C) ) ) )).

fof(d3_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
             => k3_heyting2(A,B,C,D) = k3_xboole_0(k4_partfun1(A,B),a_4_0_heyting2(A,B,C,D)) ) ) ) )).

fof(t20_heyting2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
             => ! [E] :
                  ~ ( r2_hidden(E,k3_heyting2(A,B,C,D))
                    & ! [F] :
                        ( ( v1_funct_1(F)
                          & m2_relset_1(F,C,D) )
                       => ~ ( E = k3_tarski(a_5_1_heyting2(A,B,C,D,F))
                            & k4_relset_1(C,D,F) = C ) ) ) ) ) ) )).

fof(fraenkel_a_3_0_heyting2,axiom,(
    ! [A,B,C,D] :
      ( ( v1_finset_1(C)
        & m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C))) )
     => ( r2_hidden(A,a_3_0_heyting2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k4_partfun1(k1_heyting2(B,C,D),C))
            & A = E
            & ! [F] :
                ( m1_subset_1(F,k4_partfun1(B,C))
               => ~ ( r2_hidden(F,D)
                    & r1_partfun1(E,F) ) ) ) ) ) )).

fof(fraenkel_a_4_0_heyting2,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_finset_1(C)
        & m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
        & m1_subset_1(E,k5_finsub_1(k4_partfun1(B,C))) )
     => ( r2_hidden(A,a_4_0_heyting2(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,k4_partfun1(D,E))
            & A = k3_tarski(a_5_0_heyting2(B,C,D,E,F))
            & k1_relat_1(F) = D ) ) ) )).

fof(fraenkel_a_5_0_heyting2,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( v1_finset_1(C)
        & m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
        & m1_subset_1(E,k5_finsub_1(k4_partfun1(B,C)))
        & m1_subset_1(F,k4_partfun1(D,E)) )
     => ( r2_hidden(A,a_5_0_heyting2(B,C,D,E,F))
      <=> ? [G] :
            ( m1_subset_1(G,k4_partfun1(B,C))
            & A = k4_xboole_0(k1_funct_1(F,G),G)
            & r2_hidden(G,D) ) ) ) )).

fof(fraenkel_a_5_1_heyting2,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( v1_finset_1(C)
        & m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
        & m1_subset_1(E,k5_finsub_1(k4_partfun1(B,C)))
        & v1_funct_1(F)
        & m2_relset_1(F,D,E) )
     => ( r2_hidden(A,a_5_1_heyting2(B,C,D,E,F))
      <=> ? [G] :
            ( m1_subset_1(G,k4_partfun1(B,C))
            & A = k4_xboole_0(k1_funct_1(F,G),G)
            & r2_hidden(G,D) ) ) ) )).
%------------------------------------------------------------------------------