
theorem
  for X, Y, x being set, S being Subset-Family of Y, f being
  Function of X, Y st x in meet (("f).:S) holds f.x in meet S
proof
  let X, Y, x be set, S be Subset-Family of Y, f be Function of X, Y;
  assume
A1: x in meet (("f).:S);
A2: now
    let SS be set;
    assume
A3: SS in S;
    then ("f).SS in ("f).:S by FUNCT_2:35;
    then
A4: x in ("f).SS by A1,SETFAM_1:def 1;
    ("f).SS = f"SS by A3,Def2;
    hence f.x in SS by A4,FUNCT_1:def 7;
  end;
  ("f).:S is non empty by A1,SETFAM_1:def 1;
  then S is non empty;
  hence thesis by A2,SETFAM_1:def 1;
end;
