
theorem Th86:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for v being Vertex of G1 st v in dom F_V
  holds F_E.:(v.edgesInOut()) c= (F_V/.v).edgesInOut()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  let v be Vertex of G1;
  assume A1: v in dom F_V;
  now
    let e be object;
    assume e in F_E.:(v.edgesInOut());
    then consider e0 being object such that
      A2: e0 in dom F_E & e0 in v.edgesInOut() & e = F_E.e0 by FUNCT_1:def 6;
    per cases by A2, GLIB_000:61;
    suppose A3: (the_Source_of G1).e0 = v;
      set w = (the_Target_of G1).e0;
      A4: w in dom F_V by A2, GLIB_010:5;
      e0 Joins v,w,G1 by A2, A3, GLIB_000:def 13;
      then F_E.e0 Joins F_V.v,F_V.w,G2 by A1, A2, A4, GLIB_010:4;
      then F_E.e0 Joins F_V/.v,F_V.w,G2 by A1, PARTFUN1:def 6;
      hence e in (F_V/.v).edgesInOut() by A2, GLIB_000:62;
    end;
    suppose A5: (the_Target_of G1).e0 = v;
      set w = (the_Source_of G1).e0;
      A6: w in dom F_V by A2, GLIB_010:5;
      e0 Joins v,w,G1 by A2, A5, GLIB_000:def 13;
      then F_E.e0 Joins F_V.v,F_V.w,G2 by A1, A2, A6, GLIB_010:4;
      then F_E.e0 Joins F_V/.v,F_V.w,G2 by A1, PARTFUN1:def 6;
      hence e in (F_V/.v).edgesInOut() by A2, GLIB_000:62;
    end;
  end;
  hence thesis by TARSKI:def 3;
end;
