
theorem
  for G1, G2 being _trivial loopless _Graph
  for F being non empty PGraphMapping of G1, G2
  holds F is Disomorphism &
    F = [ (the Vertex of G1) .--> the Vertex of G2, {} ]
proof
  let G1, G2 be _trivial loopless _Graph;
  let F be non empty PGraphMapping of G1, G2;
  thus F is Disomorphism;
  A1: F_E = {};
  F is total onto;
  then A2: dom F_V = the_Vertices_of G1 & rng F_V = the_Vertices_of G2;
  consider v1 being Vertex of G1 such that
    A3: the_Vertices_of G1 = {v1} by GLIB_000:22;
  consider v2 being Vertex of G2 such that
    A4: the_Vertices_of G2 = {v2} by GLIB_000:22;
  dom F_V = {the Vertex of G1} & rng F_V = {the Vertex of G2}
    by A2, A3, A4, TARSKI:def 1;
  then F_V = (the Vertex of G1) .--> the Vertex of G2 by FUNCT_4:112;
  hence thesis by A1;
end;
