
theorem
  for G1, G2 being _Graph
  for f being PartFunc of the_Vertices_of G1, the_Vertices_of G2
  holds f is directed PVertexMapping of G1, G2 iff
    for v,w,e being object st v in dom f & w in dom f & e DJoins v,w,G1
    ex e9 being object st e9 DJoins f.v,f.w,G2
proof
  let G1, G2 be _Graph;
  let f be PartFunc of the_Vertices_of G1, the_Vertices_of G2;
  thus f is directed PVertexMapping of G1, G2
    implies for v,w,e being object st v in dom f & w in dom f & e DJoins v,w,G1
      ex e9 being object st e9 DJoins f.v,f.w,G2 by Def2;
  assume A1: for v,w,e being object
    st v in dom f & w in dom f & e DJoins v,w,G1
    ex e9 being object st e9 DJoins f.v,f.w,G2;
  now
    let v,w,e be object;
    assume A2: v in dom f & w in dom f & e Joins v,w,G1;
    then per cases by GLIB_000:16;
    suppose e DJoins v,w,G1;
      then consider e9 being object such that
        A3: e9 DJoins f.v,f.w,G2 by A1, A2;
      take e9;
      thus e9 Joins f.v,f.w,G2 by A3, GLIB_000:16;
    end;
    suppose e DJoins w,v,G1;
      then consider e9 being object such that
        A4: e9 DJoins f.w,f.v,G2 by A1, A2;
      take e9;
      thus e9 Joins f.v,f.w,G2 by A4, GLIB_000:16;
    end;
  end;
  hence thesis by A1, Th1, Def2;
end;
