
theorem
  for T,S being non empty TopSpace holds for f being Function of T,S
  holds for Q being Subset-Family of S holds Q is finite implies f"Q is finite
proof
  let T,S be non empty TopSpace;
  let f be Function of T,S;
  let Q be Subset-Family of S;
  defpred EF[Subset of [#](S),Subset of [#](T)] means for s,t being set holds
  ($1 = s & $2 = t implies t = f"s);
  assume Q is finite;
  then consider s being FinSequence such that
A1: rng s = Q by FINSEQ_1:52;
A2: for x being Subset of [#](S) ex y being Subset of [#](T) st EF[x,y]
  proof
    let x be Subset of [#](S);
    reconsider x as set;
    set y = f"x;
    reconsider y as Subset of [#](T);
    take y;
    thus thesis;
  end;
  consider F being Function of bool [#](S),bool [#](T) such that
A3: for x being Subset of [#](S) holds EF[x,F.x qua Subset of [#](T)]
  from FUNCT_2:sch 3 (A2);
  dom F = bool [#](S) by FUNCT_2:def 1;
  then reconsider q = F*s as FinSequence by A1,FINSEQ_1:16;
  for x being object holds x in F.:Q iff x in f"Q
  proof
    let x be object;
A4: dom F = bool [#](S) by FUNCT_2:def 1;
    thus x in F.:Q implies x in f"Q
    proof
      assume x in F.:Q;
      then consider y being object such that
A5:   y in dom F and
A6:   y in Q & x = F.y by FUNCT_1:def 6;
      reconsider y as Subset of S by A5;
      F.y = f"y by A3;
      hence thesis by A6,FUNCT_2:def 9;
    end;
    assume
A7: x in f"Q;
    then reconsider x as Subset of T;
    consider y being Subset of S such that
A8: y in Q and
A9: x = f"y by A7,FUNCT_2:def 9;
    x = F.y by A3,A9;
    hence thesis by A8,A4,FUNCT_1:def 6;
  end;
  then
A10: F.:Q = f"Q by TARSKI:2;
  ex q being FinSequence st rng q = f"Q
  proof
    take q;
    thus thesis by A1,A10,RELAT_1:127;
  end;
  hence thesis;
end;
