reserve X1,X2,X3,X4 for set;

theorem
  for X,Y being set,f being Function of X,Y,
      SF1,SF2 being Subset-Family of X st SF1 c= SF2
  holds f.:SF1 c= f.:SF2
  proof
    let X,Y be set,f be Function of X,Y,
        SF1,SF2 be Subset-Family of X;
    assume
A1: SF1 c= SF2;
A2: f.:SF1 = {f.:A where A is Subset of X:A in SF1} &
    f.:SF2 = {f.:A where A is Subset of X: A in SF2} by Thm07;
    let x be object;
    assume x in f.:SF1;
    then consider A be Subset of X such that
A3: x=f.:A and
A4: A in SF1 by A2;
    thus x in f.:SF2 by A1,A2,A3,A4;
  end;
