reserve B for non empty set,
  A,X,x for set;

theorem Th3:
  for X,Y being set st Funcs(X,Y) <> {} & X c= A & Y c= B for f
  being Element of Funcs(X,Y) holds f is PartFunc of A,B
proof
  let X,Y be set such that
A1: Funcs(X,Y) <> {} and
A2: X c= A and
A3: Y c= B;
  let f be Element of Funcs(X,Y);
  consider g being Function such that
A4: f = g and
A5: dom g = X and
A6: rng g c= Y by A1,FUNCT_2:def 2;
  rng g c= B by A3,A6;
  hence thesis by A2,A4,A5,RELSET_1:4;
end;
