
theorem Th55:
  for G1, G2 be _Graph, F being non empty PGraphMapping of G1, G2
  holds F is total iff dom F == G1
proof
  let G1, G2 be _Graph, F be non empty PGraphMapping of G1, G2;
  hereby
    A1: G1 is Subgraph of G1 by GLIB_000:40;
    assume A2: F is total;
    A3: the_Vertices_of dom F = the_Vertices_of G1 by A2, Th54;
    the_Edges_of dom F = the_Edges_of G1 by A2, Th54;
    hence dom F == G1 by A1, A3, GLIB_000:86;
  end;
  assume A4: dom F == G1;
  A5: dom F_V = the_Vertices_of dom F by Th54
    .= the_Vertices_of G1 by A4, GLIB_000:def 34;
  dom F_E = the_Edges_of dom F by Th54
    .= the_Edges_of G1 by A4, GLIB_000:def 34;
  hence thesis by A5;
end;
