
theorem Th86:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  holds F | dom F = F & (rng F) |` F = F
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  per cases;
  suppose A1: F is non empty;
    A2: (F | dom F)_V = F_V | dom F_V by A1, GLIB_010:54
      .= F_V;
    (F | dom F)_E = F_E | dom F_E by A1, GLIB_010:54
      .= F_E;
    then [(F | dom F)_V, (F | dom F)_E] = [F_V, F_E] by A2;
    hence F | dom F = F;
    A3: ((rng F) |` F)_V = (rng F_V) |` F_V by A1, GLIB_010:54
      .= F_V;
    ((rng F) |` F)_E = (rng F_E) |` F_E by A1, GLIB_010:54
      .= F_E;
    then [((rng F) |` F)_V, ((rng F) |` F)_E] = [F_V, F_E] by A3;
    hence thesis;
  end;
  suppose A4: F is empty;
    hence F | dom F = F by Th85;
    thus thesis by A4, Th85;
  end;
end;
