%------------------------------------------------------------------------------
% File     : SET649^3 : TPTP v9.0.0. Released v3.6.0.
% Domain   : Set Theory
% Problem  : Domain R a subset of X & range R a subset of Y => R is (X to Y)
% Version  : [BS+08] axioms.
% English  : If the domain of a relation R from X to Y is a subset of X and
%            the range of R is a subset of Y, R is a relation from X to Y.

% Refs     : [BS+05] Benzmueller et al. (2005), Can a Higher-Order and a Fi
%          : [BS+08] Benzmueller et al. (2008), Combined Reasoning by Autom
%          : [Ben08] Benzmueller (2008), Email to Geoff Sutcliffe
% Source   : [Ben08]
% Names    :

% Status   : Theorem
% Rating   : 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v6.1.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v4.1.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   71 (  35 unt;  35 typ;  35 def)
%            Number of atoms       :   92 (  43 equ;   0 cnn)
%            Maximal formula atoms :    6 (   2 avg)
%            Number of connectives :  136 (   8   ~;   5   |;  19   &;  93   @)
%                                         (   1 <=>;  10  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   1 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :  216 ( 216   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   42 (  40 usr;   6 con; 0-4 aty)
%            Number of variables   :  110 (  80   ^;  22   !;   8   ?; 110   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : 
%------------------------------------------------------------------------------
%----Include basic set theory definitions
include('Axioms/SET008^0.ax').
%----Include definitions for relations
include('Axioms/SET008^2.ax').
%------------------------------------------------------------------------------
thf(thm,conjecture,
    ! [R: $i > $i > $o,X: $i > $o,Y: $i > $o] :
      ( ( ( subset @ ( rel_domain @ R ) @ X )
        & ( subset @ ( rel_codomain @ R ) @ Y ) )
     => ( sub_rel @ R @ ( cartesian_product @ X @ Y ) ) ) ).

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