
theorem Th81:
  for G1, G2 be _Graph, F being non empty one-to-one PGraphMapping of G1, G2
  holds dom F = rng(F") & rng F = dom(F")
proof
  let G1, G2 be _Graph, F be non empty one-to-one PGraphMapping of G1, G2;
  A1: the_Vertices_of dom F = dom F_V by Th54
    .= rng (F_V") by FUNCT_1:33
    .= the_Vertices_of rng(F") by Th54;
  the_Edges_of dom F = dom F_E by Th54
    .= rng (F_E") by FUNCT_1:33
    .= the_Edges_of rng(F") by Th54;
  hence dom F = rng(F") by A1, GLIB_000:86, GLIB_009:44;
  A2: the_Vertices_of rng F = rng F_V by Th54
    .= dom (F_V") by FUNCT_1:33
    .= the_Vertices_of dom(F") by Th54;
  the_Edges_of rng F = rng F_E by Th54
    .= dom (F_E") by FUNCT_1:33
    .= the_Edges_of dom(F") by Th54;
  hence rng F = dom(F") by A2, GLIB_000:86, GLIB_009:44;
end;
