reserve x,x1,x2,y,y9,y1,y2,z,z1,z2 for object,P,X,X1,X2,Y,Y1,Y2,V,Z for set;

theorem Th24:
  for f being Function holds x in dom <:f,X,Y:> iff x in dom f & x
  in X & f.x in Y
proof
  let f be Function;
  thus x in dom <:f,X,Y:> implies x in dom f & x in X & f.x in Y
  proof
    assume
A1: x in dom <:f,X,Y:>;
    then x in dom(Y|`f) /\ X by RELAT_1:61;
    then x in dom(Y|`f) by XBOOLE_0:def 4;
    hence thesis by A1,FUNCT_1:54;
  end;
  assume that
A2: x in dom f and
A3: x in X and
A4: f.x in Y;
  x in dom(Y|`f) by A2,A4,FUNCT_1:54;
  then x in dom(Y|`f) /\ X by A3,XBOOLE_0:def 4;
  hence thesis by RELAT_1:61;
end;
