%------------------------------------------------------------------------------
% File     : SET006+1 : TPTP v9.0.0. Bugfixed v2.2.1.
% Domain   : Set Theory
% Axioms   : Mapping axioms for the SET006+0 set theory axioms
% Version  : [Pas99] axioms.
% English  :

% Refs     : [Pas99] Pastre (1999), Email to G. Sutcliffe
% Source   : [Pas99]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :   17 (   0 unt;   0 def)
%            Number of atoms       :   99 (   3 equ)
%            Maximal formula atoms :   11 (   5 avg)
%            Number of connectives :   82 (   0   ~;   0   |;  46   &)
%                                         (  20 <=>;  16  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (  11 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :   14 (  13 usr;   0 prp; 2-6 aty)
%            Number of functors    :    6 (   6 usr;   0 con; 2-5 aty)
%            Number of variables   :  105 (  97   !;   8   ?)
% SPC      : 

% Comments : Requires SET006+0.ax
% Bugfixes : v2.2.1 - compose_function and inverse_function fixed.
%------------------------------------------------------------------------------
%----Axiom and properties of mappings
fof(maps,axiom,
    ! [F,A,B] :
      ( maps(F,A,B)
    <=> ( ! [X] :
            ( member(X,A)
           => ? [Y] :
                ( member(Y,B)
                & apply(F,X,Y) ) )
        & ! [X,Y1,Y2] :
            ( ( member(X,A)
              & member(Y1,B)
              & member(Y2,B) )
           => ( ( apply(F,X,Y1)
                & apply(F,X,Y2) )
             => Y1 = Y2 ) ) ) ) ).

fof(compose_predicate,axiom,
    ! [H,G,F,A,B,C] :
      ( compose_predicate(H,G,F,A,B,C)
    <=> ! [X,Z] :
          ( ( member(X,A)
            & member(Z,C) )
         => ( apply(H,X,Z)
          <=> ? [Y] :
                ( member(Y,B)
                & apply(F,X,Y)
                & apply(G,Y,Z) ) ) ) ) ).

fof(compose_function,axiom,
    ! [G,F,A,B,C,X,Z] :
      ( ( member(X,A)
        & member(Z,C) )
     => ( apply(compose_function(G,F,A,B,C),X,Z)
      <=> ? [Y] :
            ( member(Y,B)
            & apply(F,X,Y)
            & apply(G,Y,Z) ) ) ) ).

fof(equal_maps,axiom,
    ! [F,G,A,B] :
      ( equal_maps(F,G,A,B)
    <=> ! [X,Y1,Y2] :
          ( ( member(X,A)
            & member(Y1,B)
            & member(Y2,B) )
         => ( ( apply(F,X,Y1)
              & apply(G,X,Y2) )
           => Y1 = Y2 ) ) ) ).

fof(identity,axiom,
    ! [F,A] :
      ( identity(F,A)
    <=> ! [X] :
          ( member(X,A)
         => apply(F,X,X) ) ) ).

fof(injective,axiom,
    ! [F,A,B] :
      ( injective(F,A,B)
    <=> ! [X1,X2,Y] :
          ( ( member(X1,A)
            & member(X2,A)
            & member(Y,B) )
         => ( ( apply(F,X1,Y)
              & apply(F,X2,Y) )
           => X1 = X2 ) ) ) ).

fof(surjective,axiom,
    ! [F,A,B] :
      ( surjective(F,A,B)
    <=> ! [Y] :
          ( member(Y,B)
         => ? [E] :
              ( member(E,A)
              & apply(F,E,Y) ) ) ) ).

fof(one_to_one,axiom,
    ! [F,A,B] :
      ( one_to_one(F,A,B)
    <=> ( injective(F,A,B)
        & surjective(F,A,B) ) ) ).

fof(inverse_predicate,axiom,
    ! [G,F,A,B] :
      ( inverse_predicate(G,F,A,B)
    <=> ! [X,Y] :
          ( ( member(X,A)
            & member(Y,B) )
         => ( apply(F,X,Y)
          <=> apply(G,Y,X) ) ) ) ).

fof(inverse_function,axiom,
    ! [F,A,B,X,Y] :
      ( ( member(X,A)
        & member(Y,B) )
     => ( apply(F,X,Y)
      <=> apply(inverse_function(F,A,B),Y,X) ) ) ).

fof(image2,axiom,
    ! [F,A,Y] :
      ( member(Y,image2(F,A))
    <=> ? [X] :
          ( member(X,A)
          & apply(F,X,Y) ) ) ).

fof(image3,axiom,
    ! [F,A,B,Y] :
      ( member(Y,image3(F,A,B))
    <=> ( member(Y,B)
        & ? [X] :
            ( member(X,A)
            & apply(F,X,Y) ) ) ) ).

fof(inverse_image2,axiom,
    ! [F,B,X] :
      ( member(X,inverse_image2(F,B))
    <=> ? [Y] :
          ( member(Y,B)
          & apply(F,X,Y) ) ) ).

fof(inverse_image3,axiom,
    ! [F,B,A,X] :
      ( member(X,inverse_image3(F,B,A))
    <=> ( member(X,A)
        & ? [Y] :
            ( member(Y,B)
            & apply(F,X,Y) ) ) ) ).

fof(increasing_function,axiom,
    ! [F,A,R,B,S] :
      ( increasing(F,A,R,B,S)
    <=> ! [X1,Y1,X2,Y2] :
          ( ( member(X1,A)
            & member(Y1,B)
            & member(X2,A)
            & member(Y2,B)
            & apply(R,X1,X2)
            & apply(F,X1,Y1)
            & apply(F,X2,Y2) )
         => apply(S,Y1,Y2) ) ) ).

fof(decreasing_function,axiom,
    ! [F,A,R,B,S] :
      ( decreasing(F,A,R,B,S)
    <=> ! [X1,Y1,X2,Y2] :
          ( ( member(X1,A)
            & member(Y1,B)
            & member(X2,A)
            & member(Y2,B)
            & apply(R,X1,X2)
            & apply(F,X1,Y1)
            & apply(F,X2,Y2) )
         => apply(S,Y2,Y1) ) ) ).

fof(isomorphism,axiom,
    ! [F,A,R,B,S] :
      ( isomorphism(F,A,R,B,S)
    <=> ( maps(F,A,B)
        & one_to_one(F,A,B)
        & ! [X1,Y1,X2,Y2] :
            ( ( member(X1,A)
              & member(Y1,B)
              & member(X2,A)
              & member(Y2,B)
              & apply(F,X1,Y1)
              & apply(F,X2,Y2) )
           => ( apply(R,X1,X2)
            <=> apply(S,Y1,Y2) ) ) ) ) ).

%------------------------------------------------------------------------------