
theorem Th83:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is isomorphism holds G2.componentSet() =
    the set of all F_V.:C where C is Element of G1.componentSet()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  set S = the set of all F_V.:C where C is Element of G1.componentSet();
  assume A1: F is isomorphism;
  now
    let x be object;
    reconsider X = x as set by TARSKI:1;
    hereby
      assume x in G2.componentSet();
      then consider v2 being Vertex of G2 such that
        A2: X = G2.reachableFrom(v2) by GLIB_002:def 8;
      the_Vertices_of G2 = rng F_V by A1, GLIB_010:def 12;
      then consider v1 being object such that
        A3: v1 in dom F_V & F_V.v1 = v2 by FUNCT_1:def 3;
      reconsider v1 as Vertex of G1 by A3;
      A4: X = F_V.:G1.reachableFrom(v1) by A1, A2, A3, Th82;
      G1.reachableFrom(v1) in G1.componentSet() by GLIB_002:def 8;
      hence x in S by A4;
    end;
    assume x in S;
    then consider C being Element of G1.componentSet() such that
      A5: x = F_V.:C;
    consider v1 being Vertex of G1 such that
      A6: C = G1.reachableFrom(v1) by GLIB_002:def 8;
    the_Vertices_of G1 = dom F_V by A1, GLIB_010:def 11;
    then F_V.v1 in rng F_V by FUNCT_1:3;
    then reconsider v2 = F_V.v1 as Vertex of G2;
    x = G2.reachableFrom(v2) by A1, A5, A6, Th82;
    hence x in G2.componentSet() by GLIB_002:def 8;
  end;
  hence thesis by TARSKI:2;
end;
