SET007 Axioms: SET007+12.ax


%------------------------------------------------------------------------------
% File     : SET007+12 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Graphs of Functions
% 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    : grfunc_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   89 (  54 unit)
%            Number of atoms       :  287 (  33 equality)
%            Maximal formula depth :   13 (   4 average)
%            Number of connectives :  203 (   5 ~  ;   1  |;  99  &)
%                                         (   7 <=>;  91 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    8 (   1 propositional; 0-2 arity)
%            Number of functors    :   15 (   1 constant; 0-2 arity)
%            Number of variables   :   99 (   0 singleton;  99 !;   0 ?)
%            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_grfunc_1,axiom,(
    $true )).

fof(t2_grfunc_1,axiom,(
    $true )).

fof(t3_grfunc_1,axiom,(
    $true )).

fof(t4_grfunc_1,axiom,(
    $true )).

fof(t5_grfunc_1,axiom,(
    $true )).

fof(t6_grfunc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r1_tarski(B,A)
         => ( v1_relat_1(B)
            & v1_funct_1(B) ) ) ) )).

fof(t7_grfunc_1,axiom,(
    $true )).

fof(t8_grfunc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_tarski(A,B)
          <=> ( r1_tarski(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) ) ) ) ) ) )).

fof(t9_grfunc_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)
              & r1_tarski(A,B) )
           => A = B ) ) ) )).

fof(t10_grfunc_1,axiom,(
    $true )).

fof(t11_grfunc_1,axiom,(
    $true )).

fof(t12_grfunc_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ! [D] :
          ( ( v1_relat_1(D)
            & v1_funct_1(D) )
         => ( r2_hidden(k4_tarski(A,B),k5_relat_1(D,C))
           => ( r2_hidden(k4_tarski(A,k1_funct_1(D,A)),D)
              & r2_hidden(k4_tarski(k1_funct_1(D,A),B),C) ) ) ) ) )).

fof(t13_grfunc_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) )
             => ( r1_tarski(A,B)
               => ( r1_tarski(k5_relat_1(A,C),k5_relat_1(B,C))
                  & r1_tarski(k5_relat_1(C,A),k5_relat_1(C,B)) ) ) ) ) ) )).

fof(t14_grfunc_1,axiom,(
    $true )).

fof(t15_grfunc_1,axiom,(
    ! [A,B] :
      ( v1_relat_1(k1_tarski(k4_tarski(A,B)))
      & v1_funct_1(k1_tarski(k4_tarski(A,B))) ) )).

fof(t16_grfunc_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( C = k1_tarski(k4_tarski(A,B))
       => k1_funct_1(C,A) = B ) ) )).

fof(t17_grfunc_1,axiom,(
    $true )).

fof(t18_grfunc_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( k1_relat_1(B) = k1_tarski(A)
       => B = k1_tarski(k4_tarski(A,k1_funct_1(B,A))) ) ) )).

fof(t19_grfunc_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_relat_1(k2_tarski(k4_tarski(A,B),k4_tarski(C,D)))
        & v1_funct_1(k2_tarski(k4_tarski(A,B),k4_tarski(C,D))) )
    <=> ( A = C
       => B = D ) ) )).

fof(t20_grfunc_1,axiom,(
    $true )).

fof(t21_grfunc_1,axiom,(
    $true )).

fof(t22_grfunc_1,axiom,(
    $true )).

fof(t23_grfunc_1,axiom,(
    $true )).

fof(t24_grfunc_1,axiom,(
    $true )).

fof(t25_grfunc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
      <=> ! [B,C,D] :
            ( ( r2_hidden(k4_tarski(B,D),A)
              & r2_hidden(k4_tarski(C,D),A) )
           => B = C ) ) ) )).

fof(t26_grfunc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( r1_tarski(A,B)
              & v2_funct_1(B) )
           => v2_funct_1(A) ) ) ) )).

fof(t27_grfunc_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k3_xboole_0(B,A))
        & v1_funct_1(k3_xboole_0(B,A)) ) ) )).

fof(t28_grfunc_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) )
             => ( A = k3_xboole_0(B,C)
               => ( r1_tarski(k1_relat_1(A),k3_xboole_0(k1_relat_1(B),k1_relat_1(C)))
                  & r1_tarski(k2_relat_1(A),k3_xboole_0(k2_relat_1(B),k2_relat_1(C))) ) ) ) ) ) )).

fof(t29_grfunc_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) )
             => ( ( B = k3_xboole_0(C,D)
                  & r2_hidden(A,k1_relat_1(B)) )
               => ( k1_funct_1(B,A) = k1_funct_1(C,A)
                  & k1_funct_1(B,A) = k1_funct_1(D,A) ) ) ) ) ) )).

fof(t30_grfunc_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) )
             => ( C = k3_xboole_0(A,B)
               => ( ( ~ v2_funct_1(A)
                    & ~ v2_funct_1(B) )
                  | v2_funct_1(C) ) ) ) ) ) )).

fof(t31_grfunc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
           => ( v1_relat_1(k2_xboole_0(A,B))
              & v1_funct_1(k2_xboole_0(A,B)) ) ) ) ) )).

fof(t32_grfunc_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) )
             => ( ( r1_tarski(A,B)
                  & r1_tarski(C,B) )
               => ( v1_relat_1(k2_xboole_0(A,C))
                  & v1_funct_1(k2_xboole_0(A,C)) ) ) ) ) ) )).

fof(t33_grfunc_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) )
             => ( A = k2_xboole_0(B,C)
               => ( k1_relat_1(A) = k2_xboole_0(k1_relat_1(B),k1_relat_1(C))
                  & k2_relat_1(A) = k2_xboole_0(k2_relat_1(B),k2_relat_1(C)) ) ) ) ) ) )).

fof(t34_grfunc_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) )
             => ( ( r2_hidden(A,k1_relat_1(B))
                  & C = k2_xboole_0(B,D) )
               => k1_funct_1(C,A) = k1_funct_1(B,A) ) ) ) ) )).

fof(t35_grfunc_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) )
             => ( ( r2_hidden(A,k1_relat_1(B))
                  & C = k2_xboole_0(D,B) )
               => k1_funct_1(C,A) = k1_funct_1(B,A) ) ) ) ) )).

fof(t36_grfunc_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) )
             => ~ ( r2_hidden(A,k1_relat_1(B))
                  & B = k2_xboole_0(C,D)
                  & k1_funct_1(B,A) != k1_funct_1(C,A)
                  & k1_funct_1(B,A) != k1_funct_1(D,A) ) ) ) ) )).

fof(t37_grfunc_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) )
             => ( ( v2_funct_1(A)
                  & v2_funct_1(B)
                  & C = k2_xboole_0(A,B)
                  & r1_xboole_0(k2_relat_1(A),k2_relat_1(B)) )
               => v2_funct_1(C) ) ) ) ) )).

fof(t38_grfunc_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k4_xboole_0(B,A))
        & v1_funct_1(k4_xboole_0(B,A)) ) ) )).

fof(t39_grfunc_1,axiom,(
    $true )).

fof(t40_grfunc_1,axiom,(
    $true )).

fof(t41_grfunc_1,axiom,(
    $true )).

fof(t42_grfunc_1,axiom,(
    $true )).

fof(t43_grfunc_1,axiom,(
    $true )).

fof(t44_grfunc_1,axiom,(
    $true )).

fof(t45_grfunc_1,axiom,(
    $true )).

fof(t46_grfunc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( A = k1_xboole_0
       => v2_funct_1(A) ) ) )).

fof(t47_grfunc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ! [B,C] :
            ( r2_hidden(k4_tarski(C,B),k2_funct_1(A))
          <=> r2_hidden(k4_tarski(B,C),A) ) ) ) )).

fof(t48_grfunc_1,axiom,(
    $true )).

fof(t49_grfunc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( A = k1_xboole_0
       => k2_funct_1(A) = k1_xboole_0 ) ) )).

fof(t50_grfunc_1,axiom,(
    $true )).

fof(t51_grfunc_1,axiom,(
    $true )).

fof(t52_grfunc_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( ( r2_hidden(A,k1_relat_1(C))
          & r2_hidden(A,B) )
      <=> r2_hidden(k4_tarski(A,k1_funct_1(C,A)),k7_relat_1(C,B)) ) ) )).

fof(t53_grfunc_1,axiom,(
    $true )).

fof(t54_grfunc_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) )
             => ( r1_tarski(k5_relat_1(C,k7_relat_1(B,A)),k5_relat_1(C,B))
                & r1_tarski(k5_relat_1(k7_relat_1(B,A),D),k5_relat_1(B,D)) ) ) ) ) )).

fof(t55_grfunc_1,axiom,(
    $true )).

fof(t56_grfunc_1,axiom,(
    $true )).

fof(t57_grfunc_1,axiom,(
    $true )).

fof(t58_grfunc_1,axiom,(
    $true )).

fof(t59_grfunc_1,axiom,(
    $true )).

fof(t60_grfunc_1,axiom,(
    $true )).

fof(t61_grfunc_1,axiom,(
    $true )).

fof(t62_grfunc_1,axiom,(
    $true )).

fof(t63_grfunc_1,axiom,(
    $true )).

fof(t64_grfunc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_tarski(A,B)
           => k7_relat_1(B,k1_relat_1(A)) = A ) ) ) )).

fof(t65_grfunc_1,axiom,(
    $true )).

fof(t66_grfunc_1,axiom,(
    $true )).

fof(t67_grfunc_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( ( r2_hidden(A,k1_relat_1(C))
          & r2_hidden(k1_funct_1(C,A),B) )
      <=> r2_hidden(k4_tarski(A,k1_funct_1(C,A)),k8_relat_1(B,C)) ) ) )).

fof(t68_grfunc_1,axiom,(
    $true )).

fof(t69_grfunc_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) )
             => ( r1_tarski(k5_relat_1(C,k8_relat_1(A,B)),k5_relat_1(C,B))
                & r1_tarski(k5_relat_1(k8_relat_1(A,B),D),k5_relat_1(B,D)) ) ) ) ) )).

fof(t70_grfunc_1,axiom,(
    $true )).

fof(t71_grfunc_1,axiom,(
    $true )).

fof(t72_grfunc_1,axiom,(
    $true )).

fof(t73_grfunc_1,axiom,(
    $true )).

fof(t74_grfunc_1,axiom,(
    $true )).

fof(t75_grfunc_1,axiom,(
    $true )).

fof(t76_grfunc_1,axiom,(
    $true )).

fof(t77_grfunc_1,axiom,(
    $true )).

fof(t78_grfunc_1,axiom,(
    $true )).

fof(t79_grfunc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( r1_tarski(A,B)
              & v2_funct_1(B) )
           => k8_relat_1(k2_relat_1(A),B) = A ) ) ) )).

fof(t80_grfunc_1,axiom,(
    $true )).

fof(t81_grfunc_1,axiom,(
    $true )).

fof(t82_grfunc_1,axiom,(
    $true )).

fof(t83_grfunc_1,axiom,(
    $true )).

fof(t84_grfunc_1,axiom,(
    $true )).

fof(t85_grfunc_1,axiom,(
    $true )).

fof(t86_grfunc_1,axiom,(
    $true )).

fof(t87_grfunc_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(A,k10_relat_1(C,B))
      <=> ( r2_hidden(k4_tarski(A,k1_funct_1(C,A)),C)
          & r2_hidden(k1_funct_1(C,A),B) ) ) ) )).

fof(t88_grfunc_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r1_tarski(A,k1_relat_1(B))
              & r1_tarski(B,C) )
           => k7_relat_1(B,A) = k7_relat_1(C,A) ) ) ) )).

fof(t89_grfunc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_hidden(B,k1_relat_1(A))
         => k7_relat_1(A,k1_tarski(B)) = k1_tarski(k4_tarski(B,k1_funct_1(A,B))) ) ) )).
%------------------------------------------------------------------------------