reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;

theorem
  for f being Function of A,B, A0 being Subset of A, B0 being Subset of
  B holds f.:A0 c= B0 iff A0 c= f"B0
proof
  let f be Function of A,B, A0 be Subset of A, B0 be Subset of B;
  thus f.:A0 c= B0 implies A0 c= f"B0
  proof
    assume f.:A0 c= B0;
    then
A1: f"(f.:A0) c= f"B0 by RELAT_1:143;
    A0 c= f"(f.:A0) by Th41;
    hence thesis by A1;
  end;
  thus A0 c= f"B0 implies f.:A0 c= B0
  proof
    assume A0 c= f"B0;
    then
A2: f.:A0 c= f.:f"B0 by RELAT_1:123;
    f.:f"B0 c= B0 by FUNCT_1:75;
    hence thesis by A2;
  end;
end;
