
theorem Th25:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is isomorphism ex f being PVertexMapping of G1, G2
    st F_V = f & f is isomorphism
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is isomorphism;
  then reconsider f = F_V as PVertexMapping of G1, G2 by Th18;
  take f;
  thus F_V = f;
  A2: dom f = the_Vertices_of G1 by A1, GLIB_010:def 11;
  hence f is total by PARTFUN1:def 2;
  thus f is one-to-one by A1;
  rng f = the_Vertices_of G2 by A1, GLIB_010:def 12;
  hence f is onto by FUNCT_2:def 3;
  let v,w be Vertex of G1;
  card G1.edgesBetween({v},{w})
     = card G2.edgesBetween(F_V.:{v},F_V.:{w}) by A1, GLIB_010:86
    .= card G2.edgesBetween(F_V.:{v},{F_V.w}) by A2, Lm1
    .= card G2.edgesBetween({f.v},{f.w}) by A2, Lm1;
  hence thesis;
end;
