
theorem Th5:
  for A,B,C being set st C = {} implies B = {} or A = {}
  for f being Function of A, Funcs(B,C) holds doms f = A --> B
proof
  let A,B,C be set;
  assume C = {} implies B = {} or A = {};
  then
A1: Funcs(B,C) = {} implies A = {} by FUNCT_2:8;
  let f be Function of A, Funcs(B,C);
  reconsider g = f as ManySortedFunction of A by A1;
  now
    let i be object;
    assume
A2: i in A;
    then
A3: g.i in Funcs(B,C) by A1,FUNCT_2:5;
    thus (doms g).i = dom(g.i) by A2,MSSUBFAM:14
      .= B by A3,FUNCT_2:92
      .= (A-->B).i by A2,FUNCOP_1:7;
  end;
  hence thesis by PBOOLE:3;
end;
