
theorem Th81:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v1 in dom F_V & v2 = F_V.v1 & F is one-to-one onto
  holds G2.reachableFrom(v2) c= F_V.:G1.reachableFrom(v1)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 in dom F_V & v2 = F_V.v1 & F is one-to-one onto;
  then reconsider F0 = F as non empty one-to-one PGraphMapping of G1, G2;
  A2: F0"_V.v2 = v1 by A1, FUNCT_1:34;
  now
    let y be object;
    assume y in G2.reachableFrom(v2);
    then consider W2 being Walk of G2 such that
      A3: W2 is_Walk_from v2,y by GLIB_002:def 5;
    W2 is F0-valued by A1, GLIB_010:122;
    then reconsider W2 as F0"-defined Walk of G2;
    F0".:W2 is_Walk_from F0"_V.v2,F0"_V.y by A3, GLIB_010:132;
    then A4: F0"_V.y in G1.reachableFrom(v1) by A2, GLIB_002:def 5;
    y is Vertex of G2 & rng F_V = the_Vertices_of G2
      by A1, A3, GLIB_001:18, GLIB_010:def 12;
    then F0_V".y in dom F_V & y = F0_V.(F0_V".y) by FUNCT_1:32;
    hence y in F_V.:G1.reachableFrom(v1) by A4, FUNCT_1:def 6;
  end;
  hence thesis by TARSKI:def 3;
end;
