TPTP Axioms File: SET006+1.ax

```%------------------------------------------------------------------------------
% File     : SET006+1 : TPTP v7.5.0. 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 unit)
%            Number of atoms       :   99 (   3 equality)
%            Maximal formula depth :   19 (  11 average)
%            Number of connectives :   82 (   0 ~  ;   0  |;  46  &)
%                                         (  20 <=>;  16 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   14 (   0 propositional; 2-6 arity)
%            Number of functors    :    6 (   0 constant; 2-5 arity)
%            Number of variables   :  105 (   0 singleton;  97 !;   8 ?)
%            Maximal term depth    :    2 (   1 average)
% SPC      :

% 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) ) ) ) ) )).

%------------------------------------------------------------------------------
```