
theorem
  for F being Graph-yielding Function
  holds the_Vertices_of rng F = rng the_Vertices_of F
proof
  let F be Graph-yielding Function;
  now
    let z be object;
    hereby
      assume z in the_Vertices_of rng F;
      then consider G1 being _Graph such that
        A1: G1 in rng F & z = the_Vertices_of G1 by GLIB_014:def 14;
      consider x being object such that
        A2: x in dom F & G1 = F.x by A1, FUNCT_1:def 3;
      consider G2 being _Graph such that
        A3: G2 = F.x & (the_Vertices_of F).x = the_Vertices_of G2 by A2, Def4;
      x in dom the_Vertices_of F by A2, Def4;
      hence z in rng the_Vertices_of F by A1, A2, A3, FUNCT_1:3;
    end;
    assume z in rng the_Vertices_of F;
    then consider x being object such that
      A4: x in dom the_Vertices_of F & z = (the_Vertices_of F).x
      by FUNCT_1:def 3;
    A5: x in dom F by A4, Def4;
    then consider G1 being _Graph such that
      A6: G1 = F.x & (the_Vertices_of F).x = the_Vertices_of G1 by Def4;
    G1 in rng F by A5, A6, FUNCT_1:3;
    hence z in the_Vertices_of rng F by A4, A6, GLIB_014:def 14;
  end;
  hence thesis by TARSKI:2;
end;
