
theorem Th30:
  for f being non-empty disjoint_valued Function st Union f is finite
  holds dom f is finite
  proof
    let f be non-empty disjoint_valued Function;
    assume Union f is finite; then
    A1: rng f is finite
      & for X be set st X in rng f holds X is finite by FINSET_1:7;
    for x,y be object st x in dom f & y in dom f & f.x = f.y holds x = y
    proof
      let x,y be object;
      assume
      A2: x in dom f & y in dom f & f.x = f.y;
      assume x <> y; then
      A3: f.x /\ f.y  = {} by PROB_2:def 2,XBOOLE_0:def 7;
      f.x in rng f by A2,FUNCT_1:3; then
      consider i be object such that
      A4: i in f.x by XBOOLE_0:def 1;
      thus contradiction by A2,A3,A4;
    end; then
    f is one-to-one;
    hence dom f is finite by A1,CARD_1:59;
  end;
