SET007 Axioms: SET007+11.ax


%------------------------------------------------------------------------------
% File     : SET007+11 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Functions and Their Basic Properties
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : funct_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  196 (  78 unit)
%            Number of atoms       :  705 ( 145 equality)
%            Maximal formula depth :   13 (   4 average)
%            Number of connectives :  526 (  17 ~  ;   0  |; 258  &)
%                                         (  28 <=>; 223 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   15 (   1 propositional; 0-2 arity)
%            Number of functors    :   24 (   5 constant; 0-2 arity)
%            Number of variables   :  308 (   0 singleton; 294 !;  14 ?)
%            Maximal term depth    :    4 (   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(rc1_funct_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A) ) )).

fof(cc1_funct_1,axiom,(
    ! [A] :
      ( v1_xboole_0(A)
     => v1_funct_1(A) ) )).

fof(fc1_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k5_relat_1(A,B))
        & v1_funct_1(k5_relat_1(A,B)) ) ) )).

fof(fc2_funct_1,axiom,(
    ! [A] :
      ( v1_relat_1(k6_relat_1(A))
      & v1_funct_1(k6_relat_1(A)) ) )).

fof(rc2_funct_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_xboole_0(A)
      & v1_funct_1(A) ) )).

fof(cc2_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_xboole_0(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A) ) ) )).

fof(rc3_funct_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A) ) )).

fof(fc3_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A) )
     => ( v1_relat_1(k4_relat_1(A))
        & v1_funct_1(k4_relat_1(A)) ) ) )).

fof(fc4_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k7_relat_1(A,B))
        & v1_funct_1(k7_relat_1(A,B)) ) ) )).

fof(fc5_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k8_relat_1(A,B))
        & v1_funct_1(k8_relat_1(A,B)) ) ) )).

fof(rc4_funct_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) )).

fof(rc5_funct_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v2_relat_1(A)
      & v1_funct_1(A) ) )).

fof(fc6_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v2_relat_1(A)
        & v1_funct_1(A) )
     => v1_setfam_1(k2_relat_1(A)) ) )).

fof(d1_funct_1,axiom,(
    ! [A] :
      ( v1_funct_1(A)
    <=> ! [B,C,D] :
          ( ( r2_hidden(k4_tarski(B,C),A)
            & r2_hidden(k4_tarski(B,D),A) )
         => C = D ) ) )).

fof(t1_funct_1,axiom,(
    $true )).

fof(t2_funct_1,axiom,(
    ! [A] :
      ( ( ! [B] :
            ~ ( r2_hidden(B,A)
              & ! [C,D] : k4_tarski(C,D) != B )
        & ! [B,C,D] :
            ( ( r2_hidden(k4_tarski(B,C),A)
              & r2_hidden(k4_tarski(B,D),A) )
           => C = D ) )
     => ( v1_relat_1(A)
        & v1_funct_1(A) ) ) )).

fof(d2_funct_1,axiom,(
    $true )).

fof(d3_funct_1,axiom,(
    $true )).

fof(d4_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] :
          ( ( r2_hidden(B,k1_relat_1(A))
           => ( C = k1_funct_1(A,B)
            <=> r2_hidden(k4_tarski(B,C),A) ) )
          & ( ~ r2_hidden(B,k1_relat_1(A))
           => ( C = k1_funct_1(A,B)
            <=> C = k1_xboole_0 ) ) ) ) )).

fof(t3_funct_1,axiom,(
    $true )).

fof(t4_funct_1,axiom,(
    $true )).

fof(t5_funct_1,axiom,(
    $true )).

fof(t6_funct_1,axiom,(
    $true )).

fof(t7_funct_1,axiom,(
    $true )).

fof(t8_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(k4_tarski(A,B),C)
      <=> ( r2_hidden(A,k1_relat_1(C))
          & B = k1_funct_1(C,A) ) ) ) )).

fof(t9_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( k1_relat_1(A) = k1_relat_1(B)
              & ! [C] :
                  ( r2_hidden(C,k1_relat_1(A))
                 => k1_funct_1(A,C) = k1_funct_1(B,C) ) )
           => A = B ) ) ) )).

fof(d5_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( B = k2_relat_1(A)
        <=> ! [C] :
              ( r2_hidden(C,B)
            <=> ? [D] :
                  ( r2_hidden(D,k1_relat_1(A))
                  & C = k1_funct_1(A,D) ) ) ) ) )).

fof(t10_funct_1,axiom,(
    $true )).

fof(t11_funct_1,axiom,(
    $true )).

fof(t12_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k1_relat_1(B))
       => r2_hidden(k1_funct_1(B,A),k2_relat_1(B)) ) ) )).

fof(t13_funct_1,axiom,(
    $true )).

fof(t14_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( k1_relat_1(B) = k1_tarski(A)
       => k2_relat_1(B) = k1_tarski(k1_funct_1(B,A)) ) ) )).

fof(t15_funct_1,axiom,(
    ! [A] :
      ( A != k1_xboole_0
     => ! [B] :
        ? [C] :
          ( v1_relat_1(C)
          & v1_funct_1(C)
          & k1_relat_1(C) = A
          & k2_relat_1(C) = k1_tarski(B) ) ) )).

fof(t16_funct_1,axiom,(
    ! [A] :
      ( ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( ( k1_relat_1(B) = A
                  & k1_relat_1(C) = A )
               => B = C ) ) )
     => A = k1_xboole_0 ) )).

fof(t17_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( k1_relat_1(B) = k1_relat_1(C)
              & k2_relat_1(B) = k1_tarski(A)
              & k2_relat_1(C) = k1_tarski(A) )
           => B = C ) ) ) )).

fof(t18_funct_1,axiom,(
    ! [A,B] :
      ~ ( ~ ( A = k1_xboole_0
            & B != k1_xboole_0 )
        & ! [C] :
            ( ( v1_relat_1(C)
              & v1_funct_1(C) )
           => ~ ( B = k1_relat_1(C)
                & r1_tarski(k2_relat_1(C),A) ) ) ) )).

fof(t19_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( ! [C] :
            ~ ( r2_hidden(C,A)
              & ! [D] :
                  ~ ( r2_hidden(D,k1_relat_1(B))
                    & C = k1_funct_1(B,D) ) )
       => r1_tarski(A,k2_relat_1(B)) ) ) )).

fof(t20_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( ( ! [D] :
                      ( r2_hidden(D,k1_relat_1(C))
                    <=> ( r2_hidden(D,k1_relat_1(A))
                        & r2_hidden(k1_funct_1(A,D),k1_relat_1(B)) ) )
                  & ! [D] :
                      ( r2_hidden(D,k1_relat_1(C))
                     => k1_funct_1(C,D) = k1_funct_1(B,k1_funct_1(A,D)) ) )
               => C = k5_relat_1(A,B) ) ) ) ) )).

fof(t21_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(k5_relat_1(C,B)))
          <=> ( r2_hidden(A,k1_relat_1(C))
              & r2_hidden(k1_funct_1(C,A),k1_relat_1(B)) ) ) ) ) )).

fof(t22_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(k5_relat_1(C,B)))
           => k1_funct_1(k5_relat_1(C,B),A) = k1_funct_1(B,k1_funct_1(C,A)) ) ) ) )).

fof(t23_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(B))
           => k1_funct_1(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) )).

fof(t24_funct_1,axiom,(
    $true )).

fof(t25_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k2_relat_1(k5_relat_1(C,B)))
           => r2_hidden(A,k2_relat_1(B)) ) ) ) )).

fof(t26_funct_1,axiom,(
    $true )).

fof(t27_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( k1_relat_1(k5_relat_1(B,A)) = k1_relat_1(B)
           => r1_tarski(k2_relat_1(B),k1_relat_1(A)) ) ) ) )).

fof(t28_funct_1,axiom,(
    $true )).

fof(t29_funct_1,axiom,(
    $true )).

fof(t30_funct_1,axiom,(
    $true )).

fof(t31_funct_1,axiom,(
    $true )).

fof(t32_funct_1,axiom,(
    $true )).

fof(t33_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( ( r1_tarski(k2_relat_1(B),A)
          & ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D) )
                 => ( ( k1_relat_1(C) = A
                      & k1_relat_1(D) = A
                      & k5_relat_1(B,C) = k5_relat_1(B,D) )
                   => C = D ) ) ) )
       => A = k2_relat_1(B) ) ) )).

fof(t34_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( B = k6_relat_1(A)
      <=> ( k1_relat_1(B) = A
          & ! [C] :
              ( r2_hidden(C,A)
             => k1_funct_1(B,C) = C ) ) ) ) )).

fof(t35_funct_1,axiom,(
    ! [A,B] :
      ( r2_hidden(B,A)
     => k1_funct_1(k6_relat_1(A),B) = B ) )).

fof(t36_funct_1,axiom,(
    $true )).

fof(t37_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => k1_relat_1(k5_relat_1(k6_relat_1(A),B)) = k3_xboole_0(k1_relat_1(B),A) ) )).

fof(t38_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,k3_xboole_0(k1_relat_1(C),A))
       => k1_funct_1(C,B) = k1_funct_1(k5_relat_1(k6_relat_1(A),C),B) ) ) )).

fof(t39_funct_1,axiom,(
    $true )).

fof(t40_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,k1_relat_1(k5_relat_1(C,k6_relat_1(A))))
      <=> ( r2_hidden(B,k1_relat_1(C))
          & r2_hidden(k1_funct_1(C,B),A) ) ) ) )).

fof(t41_funct_1,axiom,(
    $true )).

fof(t42_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( k5_relat_1(k6_relat_1(k1_relat_1(A)),A) = A
        & k5_relat_1(A,k6_relat_1(k2_relat_1(A))) = A ) ) )).

fof(t43_funct_1,axiom,(
    ! [A,B] : k5_relat_1(k6_relat_1(B),k6_relat_1(A)) = k6_relat_1(k3_xboole_0(A,B)) )).

fof(t44_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( k2_relat_1(A) = k1_relat_1(B)
              & k5_relat_1(A,B) = A )
           => B = k6_relat_1(k1_relat_1(B)) ) ) ) )).

fof(d6_funct_1,axiom,(
    $true )).

fof(d7_funct_1,axiom,(
    $true )).

fof(d8_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
      <=> ! [B,C] :
            ( ( r2_hidden(B,k1_relat_1(A))
              & r2_hidden(C,k1_relat_1(A))
              & k1_funct_1(A,B) = k1_funct_1(A,C) )
           => B = C ) ) ) )).

fof(t45_funct_1,axiom,(
    $true )).

fof(t46_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( v2_funct_1(A)
              & v2_funct_1(B) )
           => v2_funct_1(k5_relat_1(A,B)) ) ) ) )).

fof(t47_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( v2_funct_1(k5_relat_1(B,A))
              & r1_tarski(k2_relat_1(B),k1_relat_1(A)) )
           => v2_funct_1(B) ) ) ) )).

fof(t48_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( v2_funct_1(k5_relat_1(B,A))
              & k2_relat_1(B) = k1_relat_1(A) )
           => ( v2_funct_1(B)
              & v2_funct_1(A) ) ) ) ) )).

fof(t49_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
      <=> ! [B] :
            ( ( v1_relat_1(B)
              & v1_funct_1(B) )
           => ! [C] :
                ( ( v1_relat_1(C)
                  & v1_funct_1(C) )
               => ( ( r1_tarski(k2_relat_1(B),k1_relat_1(A))
                    & r1_tarski(k2_relat_1(C),k1_relat_1(A))
                    & k1_relat_1(B) = k1_relat_1(C)
                    & k5_relat_1(B,A) = k5_relat_1(C,A) )
                 => B = C ) ) ) ) ) )).

fof(t50_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( k1_relat_1(B) = A
              & k1_relat_1(C) = A
              & r1_tarski(k2_relat_1(C),A)
              & v2_funct_1(B)
              & k5_relat_1(C,B) = B )
           => C = k6_relat_1(A) ) ) ) )).

fof(t51_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( k2_relat_1(k5_relat_1(B,A)) = k2_relat_1(A)
              & v2_funct_1(A) )
           => r1_tarski(k1_relat_1(A),k2_relat_1(B)) ) ) ) )).

fof(t52_funct_1,axiom,(
    ! [A] : v2_funct_1(k6_relat_1(A)) )).

fof(t53_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ? [B] :
            ( v1_relat_1(B)
            & v1_funct_1(B)
            & k5_relat_1(A,B) = k6_relat_1(k1_relat_1(A)) )
       => v2_funct_1(A) ) ) )).

fof(d9_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => k2_funct_1(A) = k4_relat_1(A) ) ) )).

fof(t54_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ! [B] :
            ( ( v1_relat_1(B)
              & v1_funct_1(B) )
           => ( B = k2_funct_1(A)
            <=> ( k1_relat_1(B) = k2_relat_1(A)
                & ! [C,D] :
                    ( ( ( r2_hidden(C,k2_relat_1(A))
                        & D = k1_funct_1(B,C) )
                     => ( r2_hidden(D,k1_relat_1(A))
                        & C = k1_funct_1(A,D) ) )
                    & ( ( r2_hidden(D,k1_relat_1(A))
                        & C = k1_funct_1(A,D) )
                     => ( r2_hidden(C,k2_relat_1(A))
                        & D = k1_funct_1(B,C) ) ) ) ) ) ) ) ) )).

fof(t55_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ( k2_relat_1(A) = k1_relat_1(k2_funct_1(A))
          & k1_relat_1(A) = k2_relat_1(k2_funct_1(A)) ) ) ) )).

fof(t56_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( ( v2_funct_1(B)
          & r2_hidden(A,k1_relat_1(B)) )
       => ( A = k1_funct_1(k2_funct_1(B),k1_funct_1(B,A))
          & A = k1_funct_1(k5_relat_1(B,k2_funct_1(B)),A) ) ) ) )).

fof(t57_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( ( v2_funct_1(B)
          & r2_hidden(A,k2_relat_1(B)) )
       => ( A = k1_funct_1(B,k1_funct_1(k2_funct_1(B),A))
          & A = k1_funct_1(k5_relat_1(k2_funct_1(B),B),A) ) ) ) )).

fof(t58_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ( k1_relat_1(k5_relat_1(A,k2_funct_1(A))) = k1_relat_1(A)
          & k2_relat_1(k5_relat_1(A,k2_funct_1(A))) = k1_relat_1(A) ) ) ) )).

fof(t59_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ( k1_relat_1(k5_relat_1(k2_funct_1(A),A)) = k2_relat_1(A)
          & k2_relat_1(k5_relat_1(k2_funct_1(A),A)) = k2_relat_1(A) ) ) ) )).

fof(t60_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( v2_funct_1(A)
              & k1_relat_1(A) = k2_relat_1(B)
              & k2_relat_1(A) = k1_relat_1(B)
              & ! [C,D] :
                  ( ( r2_hidden(C,k1_relat_1(A))
                    & r2_hidden(D,k1_relat_1(B)) )
                 => ( k1_funct_1(A,C) = D
                  <=> k1_funct_1(B,D) = C ) ) )
           => B = k2_funct_1(A) ) ) ) )).

fof(t61_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ( k5_relat_1(A,k2_funct_1(A)) = k6_relat_1(k1_relat_1(A))
          & k5_relat_1(k2_funct_1(A),A) = k6_relat_1(k2_relat_1(A)) ) ) ) )).

fof(t62_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => v2_funct_1(k2_funct_1(A)) ) ) )).

fof(t63_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( v2_funct_1(A)
              & k2_relat_1(A) = k1_relat_1(B)
              & k5_relat_1(A,B) = k6_relat_1(k1_relat_1(A)) )
           => B = k2_funct_1(A) ) ) ) )).

fof(t64_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( v2_funct_1(A)
              & k2_relat_1(B) = k1_relat_1(A)
              & k5_relat_1(B,A) = k6_relat_1(k2_relat_1(A)) )
           => B = k2_funct_1(A) ) ) ) )).

fof(t65_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => k2_funct_1(k2_funct_1(A)) = A ) ) )).

fof(t66_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( v2_funct_1(A)
              & v2_funct_1(B) )
           => k2_funct_1(k5_relat_1(A,B)) = k5_relat_1(k2_funct_1(B),k2_funct_1(A)) ) ) ) )).

fof(t67_funct_1,axiom,(
    ! [A] : k2_funct_1(k6_relat_1(A)) = k6_relat_1(A) )).

fof(t68_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( B = k7_relat_1(C,A)
          <=> ( k1_relat_1(B) = k3_xboole_0(k1_relat_1(C),A)
              & ! [D] :
                  ( r2_hidden(D,k1_relat_1(B))
                 => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) )).

fof(t69_funct_1,axiom,(
    $true )).

fof(t70_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,k1_relat_1(k7_relat_1(C,A)))
       => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) )).

fof(t71_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,k3_xboole_0(k1_relat_1(C),A))
       => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) )).

fof(t72_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,A)
       => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) )).

fof(t73_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( ( r2_hidden(B,k1_relat_1(C))
          & r2_hidden(B,A) )
       => r2_hidden(k1_funct_1(C,B),k2_relat_1(k7_relat_1(C,A))) ) ) )).

fof(t74_funct_1,axiom,(
    $true )).

fof(t75_funct_1,axiom,(
    $true )).

fof(t76_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_tarski(k1_relat_1(k7_relat_1(B,A)),k1_relat_1(B))
        & r1_tarski(k2_relat_1(k7_relat_1(B,A)),k2_relat_1(B)) ) ) )).

fof(t77_funct_1,axiom,(
    $true )).

fof(t78_funct_1,axiom,(
    $true )).

fof(t79_funct_1,axiom,(
    $true )).

fof(t80_funct_1,axiom,(
    $true )).

fof(t81_funct_1,axiom,(
    $true )).

fof(t82_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r1_tarski(A,B)
       => ( k7_relat_1(k7_relat_1(C,A),B) = k7_relat_1(C,A)
          & k7_relat_1(k7_relat_1(C,B),A) = k7_relat_1(C,A) ) ) ) )).

fof(t83_funct_1,axiom,(
    $true )).

fof(t84_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v2_funct_1(B)
       => v2_funct_1(k7_relat_1(B,A)) ) ) )).

fof(t85_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( B = k8_relat_1(A,C)
          <=> ( ! [D] :
                  ( r2_hidden(D,k1_relat_1(B))
                <=> ( r2_hidden(D,k1_relat_1(C))
                    & r2_hidden(k1_funct_1(C,D),A) ) )
              & ! [D] :
                  ( r2_hidden(D,k1_relat_1(B))
                 => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) )).

fof(t86_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,k1_relat_1(k8_relat_1(A,C)))
      <=> ( r2_hidden(B,k1_relat_1(C))
          & r2_hidden(k1_funct_1(C,B),A) ) ) ) )).

fof(t87_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,k1_relat_1(k8_relat_1(A,C)))
       => k1_funct_1(k8_relat_1(A,C),B) = k1_funct_1(C,B) ) ) )).

fof(t88_funct_1,axiom,(
    $true )).

fof(t89_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_tarski(k1_relat_1(k8_relat_1(A,B)),k1_relat_1(B))
        & r1_tarski(k2_relat_1(k8_relat_1(A,B)),k2_relat_1(B)) ) ) )).

fof(t90_funct_1,axiom,(
    $true )).

fof(t91_funct_1,axiom,(
    $true )).

fof(t92_funct_1,axiom,(
    $true )).

fof(t93_funct_1,axiom,(
    $true )).

fof(t94_funct_1,axiom,(
    $true )).

fof(t95_funct_1,axiom,(
    $true )).

fof(t96_funct_1,axiom,(
    $true )).

fof(t97_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r1_tarski(A,B)
       => ( k8_relat_1(B,k8_relat_1(A,C)) = k8_relat_1(A,C)
          & k8_relat_1(A,k8_relat_1(B,C)) = k8_relat_1(A,C) ) ) ) )).

fof(t98_funct_1,axiom,(
    $true )).

fof(t99_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v2_funct_1(B)
       => v2_funct_1(k8_relat_1(A,B)) ) ) )).

fof(d10_funct_1,axiom,(
    $true )).

fof(d11_funct_1,axiom,(
    $true )).

fof(d12_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] :
          ( C = k9_relat_1(A,B)
        <=> ! [D] :
              ( r2_hidden(D,C)
            <=> ? [E] :
                  ( r2_hidden(E,k1_relat_1(A))
                  & r2_hidden(E,B)
                  & D = k1_funct_1(A,E) ) ) ) ) )).

fof(t100_funct_1,axiom,(
    $true )).

fof(t101_funct_1,axiom,(
    $true )).

fof(t102_funct_1,axiom,(
    $true )).

fof(t103_funct_1,axiom,(
    $true )).

fof(t104_funct_1,axiom,(
    $true )).

fof(t105_funct_1,axiom,(
    $true )).

fof(t106_funct_1,axiom,(
    $true )).

fof(t107_funct_1,axiom,(
    $true )).

fof(t108_funct_1,axiom,(
    $true )).

fof(t109_funct_1,axiom,(
    $true )).

fof(t110_funct_1,axiom,(
    $true )).

fof(t111_funct_1,axiom,(
    $true )).

fof(t112_funct_1,axiom,(
    $true )).

fof(t113_funct_1,axiom,(
    $true )).

fof(t114_funct_1,axiom,(
    $true )).

fof(t115_funct_1,axiom,(
    $true )).

fof(t116_funct_1,axiom,(
    $true )).

fof(t117_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k1_relat_1(B))
       => k9_relat_1(B,k1_tarski(A)) = k1_tarski(k1_funct_1(B,A)) ) ) )).

fof(t118_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( ( r2_hidden(A,k1_relat_1(C))
          & r2_hidden(B,k1_relat_1(C)) )
       => k9_relat_1(C,k2_tarski(A,B)) = k2_tarski(k1_funct_1(C,A),k1_funct_1(C,B)) ) ) )).

fof(t119_funct_1,axiom,(
    $true )).

fof(t120_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => r1_tarski(k9_relat_1(k8_relat_1(A,C),B),k9_relat_1(C,B)) ) )).

fof(t121_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( v2_funct_1(C)
       => k9_relat_1(C,k3_xboole_0(A,B)) = k3_xboole_0(k9_relat_1(C,A),k9_relat_1(C,B)) ) ) )).

fof(t122_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ! [B,C] : k9_relat_1(A,k3_xboole_0(B,C)) = k3_xboole_0(k9_relat_1(A,B),k9_relat_1(A,C))
       => v2_funct_1(A) ) ) )).

fof(t123_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( v2_funct_1(C)
       => k9_relat_1(C,k4_xboole_0(A,B)) = k4_xboole_0(k9_relat_1(C,A),k9_relat_1(C,B)) ) ) )).

fof(t124_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ! [B,C] : k9_relat_1(A,k4_xboole_0(B,C)) = k4_xboole_0(k9_relat_1(A,B),k9_relat_1(A,C))
       => v2_funct_1(A) ) ) )).

fof(t125_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( ( r1_xboole_0(A,B)
          & v2_funct_1(C) )
       => r1_xboole_0(k9_relat_1(C,A),k9_relat_1(C,B)) ) ) )).

fof(t126_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => k9_relat_1(k8_relat_1(A,C),B) = k3_xboole_0(A,k9_relat_1(C,B)) ) )).

fof(d13_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] :
          ( C = k10_relat_1(A,B)
        <=> ! [D] :
              ( r2_hidden(D,C)
            <=> ( r2_hidden(D,k1_relat_1(A))
                & r2_hidden(k1_funct_1(A,D),B) ) ) ) ) )).

fof(t127_funct_1,axiom,(
    $true )).

fof(t128_funct_1,axiom,(
    $true )).

fof(t129_funct_1,axiom,(
    $true )).

fof(t130_funct_1,axiom,(
    $true )).

fof(t131_funct_1,axiom,(
    $true )).

fof(t132_funct_1,axiom,(
    $true )).

fof(t133_funct_1,axiom,(
    $true )).

fof(t134_funct_1,axiom,(
    $true )).

fof(t135_funct_1,axiom,(
    $true )).

fof(t136_funct_1,axiom,(
    $true )).

fof(t137_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => k10_relat_1(C,k3_xboole_0(A,B)) = k3_xboole_0(k10_relat_1(C,A),k10_relat_1(C,B)) ) )).

fof(t138_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => k10_relat_1(C,k4_xboole_0(A,B)) = k4_xboole_0(k10_relat_1(C,A),k10_relat_1(C,B)) ) )).

fof(t139_funct_1,axiom,(
    ! [A,B,C] :
      ( v1_relat_1(C)
     => k10_relat_1(k7_relat_1(C,A),B) = k3_xboole_0(A,k10_relat_1(C,B)) ) )).

fof(t140_funct_1,axiom,(
    $true )).

fof(t141_funct_1,axiom,(
    $true )).

fof(t142_funct_1,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ( r2_hidden(A,k2_relat_1(B))
      <=> k10_relat_1(B,k1_tarski(A)) != k1_xboole_0 ) ) )).

fof(t143_funct_1,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ( ! [C] :
            ~ ( r2_hidden(C,A)
              & k10_relat_1(B,k1_tarski(C)) = k1_xboole_0 )
       => r1_tarski(A,k2_relat_1(B)) ) ) )).

fof(t144_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ! [B] :
            ~ ( r2_hidden(B,k2_relat_1(A))
              & ! [C] : k10_relat_1(A,k1_tarski(B)) != k1_tarski(C) )
      <=> v2_funct_1(A) ) ) )).

fof(t145_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => r1_tarski(k9_relat_1(B,k10_relat_1(B,A)),A) ) )).

fof(t146_funct_1,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ( r1_tarski(A,k1_relat_1(B))
       => r1_tarski(A,k10_relat_1(B,k9_relat_1(B,A))) ) ) )).

fof(t147_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_tarski(A,k2_relat_1(B))
       => k9_relat_1(B,k10_relat_1(B,A)) = A ) ) )).

fof(t148_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => k9_relat_1(B,k10_relat_1(B,A)) = k3_xboole_0(A,k9_relat_1(B,k1_relat_1(B))) ) )).

fof(t149_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => r1_tarski(k9_relat_1(C,k3_xboole_0(A,k10_relat_1(C,B))),k3_xboole_0(k9_relat_1(C,A),B)) ) )).

fof(t150_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => k9_relat_1(C,k3_xboole_0(A,k10_relat_1(C,B))) = k3_xboole_0(k9_relat_1(C,A),B) ) )).

fof(t151_funct_1,axiom,(
    ! [A,B,C] :
      ( v1_relat_1(C)
     => r1_tarski(k3_xboole_0(A,k10_relat_1(C,B)),k10_relat_1(C,k3_xboole_0(k9_relat_1(C,A),B))) ) )).

fof(t152_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v2_funct_1(B)
       => r1_tarski(k10_relat_1(B,k9_relat_1(B,A)),A) ) ) )).

fof(t153_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ! [B] : r1_tarski(k10_relat_1(A,k9_relat_1(A,B)),B)
       => v2_funct_1(A) ) ) )).

fof(t154_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v2_funct_1(B)
       => k9_relat_1(B,A) = k10_relat_1(k2_funct_1(B),A) ) ) )).

fof(t155_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v2_funct_1(B)
       => k10_relat_1(B,A) = k9_relat_1(k2_funct_1(B),A) ) ) )).

fof(t156_funct_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ! [D] :
              ( ( v1_relat_1(D)
                & v1_funct_1(D) )
             => ( ( A = k2_relat_1(B)
                  & k1_relat_1(C) = A
                  & k1_relat_1(D) = A
                  & k5_relat_1(B,C) = k5_relat_1(B,D) )
               => C = D ) ) ) ) )).

fof(t157_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( ( r1_tarski(k9_relat_1(C,A),k9_relat_1(C,B))
          & r1_tarski(A,k1_relat_1(C))
          & v2_funct_1(C) )
       => r1_tarski(A,B) ) ) )).

fof(t158_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( ( r1_tarski(k10_relat_1(C,A),k10_relat_1(C,B))
          & r1_tarski(A,k2_relat_1(C)) )
       => r1_tarski(A,B) ) ) )).

fof(t159_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
      <=> ! [B] :
          ? [C] : r1_tarski(k10_relat_1(A,k1_tarski(B)),k1_tarski(C)) ) ) )).

fof(t160_funct_1,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ! [C] :
          ( v1_relat_1(C)
         => ( r1_tarski(k2_relat_1(B),k1_relat_1(C))
           => r1_tarski(k10_relat_1(B,A),k10_relat_1(k5_relat_1(B,C),k9_relat_1(C,A))) ) ) ) )).

fof(t161_funct_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( ( k10_relat_1(C,A) = k10_relat_1(C,B)
          & r1_tarski(A,k2_relat_1(C))
          & r1_tarski(B,k2_relat_1(C)) )
       => A = B ) ) )).

fof(t162_funct_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => k9_relat_1(k6_relat_1(A),B) = B ) )).

fof(d14_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v3_relat_1(A)
      <=> ! [B] :
            ( r2_hidden(B,k1_relat_1(A))
           => v1_xboole_0(k1_funct_1(A,B)) ) ) ) )).

fof(d15_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_relat_1(A)
      <=> ! [B] :
            ~ ( r2_hidden(B,k1_relat_1(A))
              & v1_xboole_0(k1_funct_1(A,B)) ) ) ) )).

fof(s1_funct_1,axiom,
    ( ! [A,B,C] :
        ( ( p1_s1_funct_1(A,B)
          & p1_s1_funct_1(A,C) )
       => B = C )
   => ? [A] :
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & ! [B,C] :
            ( r2_hidden(k4_tarski(B,C),A)
          <=> ( r2_hidden(B,f1_s1_funct_1)
              & p1_s1_funct_1(B,C) ) ) ) )).

fof(s2_funct_1,axiom,
    ( ( ! [A,B,C] :
          ( ( r2_hidden(A,f1_s2_funct_1)
            & p1_s2_funct_1(A,B)
            & p1_s2_funct_1(A,C) )
         => B = C )
      & ! [A] :
          ~ ( r2_hidden(A,f1_s2_funct_1)
            & ! [B] : ~ p1_s2_funct_1(A,B) ) )
   => ? [A] :
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = f1_s2_funct_1
        & ! [B] :
            ( r2_hidden(B,f1_s2_funct_1)
           => p1_s2_funct_1(B,k1_funct_1(A,B)) ) ) )).

fof(s3_funct_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & k1_relat_1(A) = f1_s3_funct_1
      & ! [B] :
          ( r2_hidden(B,f1_s3_funct_1)
         => k1_funct_1(A,B) = f2_s3_funct_1(B) ) ) )).

fof(s4_funct_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & k1_relat_1(A) = f1_s4_funct_1
      & ! [B] :
          ( m1_subset_1(B,f1_s4_funct_1)
         => k1_funct_1(A,B) = f2_s4_funct_1(B) ) ) )).

fof(dt_k1_funct_1,axiom,(
    $true )).

fof(dt_k2_funct_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k2_funct_1(A))
        & v1_funct_1(k2_funct_1(A)) ) ) )).
%------------------------------------------------------------------------------