
theorem
  for G1, G2 be _Graph, F being non empty PGraphMapping of G1, G2
  holds F is onto iff rng F == G2
proof
  let G1, G2 be _Graph, F be non empty PGraphMapping of G1, G2;
  hereby
    A1: G2 is Subgraph of G2 by GLIB_000:40;
    assume A2: F is onto;
    A3: the_Vertices_of rng F = the_Vertices_of G2 by A2, Th54;
    the_Edges_of rng F = the_Edges_of G2 by A2, Th54;
    hence rng F == G2 by A1, A3, GLIB_000:86;
  end;
  assume A4: rng F == G2;
  A5: rng F_V = the_Vertices_of rng F by Th54
    .= the_Vertices_of G2 by A4, GLIB_000:def 34;
  rng F_E = the_Edges_of rng F by Th54
    .= the_Edges_of G2 by A4, GLIB_000:def 34;
  hence thesis by A5;
end;
