reserve A, B, X, Y, Z, x, y for set;
reserve f for Function;
reserve O for Ordinal;

theorem Th10:
  for f being Function holds dom f is finite iff f is finite
proof
  let f be Function;
  thus dom f is finite implies f is finite
  proof
    assume
A1: dom f is finite;
    then
A2: rng f is finite by Th8;
    f c= [:dom f, rng f:] by RELAT_1:7;
    hence thesis by A1,A2;
  end;
  pr1(dom f,rng f).:f = dom f by FUNCT_3:79;
  hence thesis;
end;
