
theorem Th54:
  for G1, G2 being _Graph, F being non empty PGraphMapping of G1, G2
  holds the_Vertices_of dom F = dom F_V & the_Edges_of dom F = dom F_E &
    the_Vertices_of rng F = rng F_V & the_Edges_of rng F = rng F_E
proof
  let G1, G2 be _Graph;
  let F be non empty PGraphMapping of G1, G2;
  dom F_E c= G1.edgesBetween(dom F_V) by Th7;
  hence the_Vertices_of dom F = dom F_V & the_Edges_of dom F = dom F_E
    by GLIB_000:def 37;
  rng F_E c= G2.edgesBetween(rng F_V) by Th7;
  hence the_Vertices_of rng F = rng F_V & the_Edges_of rng F = rng F_E
    by GLIB_000:def 37;
end;
