reserve X1,X2,X3,X4 for set;

theorem Thm07:
  for T,S being set, f being Function of T,S,
      G being Subset-Family of T holds
    f.:G = {f.:A where A is Subset of T : A in G}
  proof
    let T,S be set,
    f be Function of T,S,
    G be Subset-Family of T;
    hereby
      let t be object;
      assume
A1:   t in f.:G;
      then reconsider t1=t as Subset of S;
      consider B be Subset of T such that
A2:   B in G and
A3:   t1 = f.:B by A1,FUNCT_2:def 10;
      thus t in {f.:A where A is Subset of T:A in G} by A2,A3;
    end;
    let t be object;
    assume t in {f.:A where A is Subset of T:A in G};
    then consider A be Subset of T such that
A4: t = f.:A and
A5: A in G;
    thus t in f.:G by A4,A5,FUNCT_2:def 10;
  end;
