
theorem Th7:
  for X being set, f being Function holds rng (X-indexing f) = (X \
  dom f) \/ f.:X
proof
  let X be set, f be Function;
  dom id X = X;
  hence rng (X-indexing f) = (id X).:(X\dom (f|X)) \/ rng (f|X) by FRECHET:12
    .= (id X).:(X\dom (f|X)) \/ f.:X by RELAT_1:115
    .= (X\dom (f|X)) \/ f.:X by FUNCT_1:92
    .= (X\(dom f /\ X)) \/ f.:X by RELAT_1:61
    .= (X \ dom f) \/ f.:X by XBOOLE_1:47;
end;
