
theorem Th87:
  for G being _Graph, E1, E2 being RepDEdgeSelection of G
  ex f being one-to-one Function st dom f = E1 & rng f = E2 &
    for e,v,w being object st e in E1 holds e DJoins v,w,G iff f.e DJoins v,w,G
proof
  let G be _Graph, E1, E2 be RepDEdgeSelection of G;
  defpred P[object,object] means $2 in E2 & ex v,w being object
    st $1 DJoins v,w,G & $2 DJoins v,w,G;
  A1: for x,y1,y2 being object st x in E1 & P[x,y1] & P[x,y2] holds y1 = y2
  proof
    let x,y1,y2 be object;
    assume A2: x in E1 & P[x,y1] & P[x,y2];
    then consider v1,w1 being object such that
      A3: x DJoins v1,w1,G & y1 DJoins v1,w1,G;
    consider v2,w2 being object such that
      A4: x DJoins v2,w2,G & y2 DJoins v2,w2,G by A2;
    consider e2 being object such that
      e2 DJoins v1,w1,G & e2 in E2 and
      A5: for e9 being object st e9 DJoins v1,w1,G & e9 in E2 holds e9 = e2
      by A3, Def6;
    A6: y1 = e2 by A2, A3, A5;
    v1 = v2 & w1 = w2 by A3, A4, GLIB_000:125;
    hence y1 = y2 by A2, A4, A5, A6;
  end;
  A7: for x being object st x in E1 ex y being object st P[x,y]
  proof
    let x be object;
    set v = (the_Source_of G).x, w = (the_Target_of G).x;
    assume A8: x in E1;
    then consider e2 being object such that
      A9: e2 DJoins v,w,G & e2 in E2 and
      for e9 being object st e9 DJoins v,w,G & e9 in E2 holds e9 = e2
      by Def6, GLIB_000:def 14;
    take e2;
    thus e2 in E2 by A9;
    take v,w;
    thus thesis by A8, A9, GLIB_000:def 14;
  end;
  consider f being Function such that
    A10: dom f = E1 & for x being object st x in E1 holds P[x,f.x]
    from FUNCT_1:sch 2(A1,A7);
  now
    let x1,x2 be object;
    assume A11: x1 in dom f & x2 in dom f & f.x1 = f.x2;
    then consider v1,w1 being object such that
      A12: x1 DJoins v1,w1,G & f.x1 DJoins v1,w1,G by A10;
    consider v2,w2 being object such that
      A13: x2 DJoins v2,w2,G & f.x2 DJoins v2,w2,G by A10, A11;
    consider e1 being object such that
      e1 DJoins v1,w1,G & e1 in E1 and
      A14: for e9 being object st e9 DJoins v1,w1,G & e9 in E1 holds e9 = e1
      by A12, Def6;
    A15: x1 = e1 by A10, A11, A12, A14;
    v1 = v2 & w1 = w2 by A11, A12, A13, GLIB_000:125;
    hence x1 = x2 by A10, A11, A13, A14, A15;
  end;
  then reconsider f as one-to-one Function by FUNCT_1:def 4;
  take f;
  thus dom f = E1 by A10;
  now
    let y be object;
    hereby
      assume y in rng f;
      then consider x being object such that
        A16: x in dom f & f.x = y by FUNCT_1:def 3;
      thus y in E2 by A10, A16;
    end;
    assume A17: y in E2;
    set v = (the_Source_of G).y, w = (the_Target_of G).y;
    consider e1 being object such that
      A18: e1 DJoins v,w,G & e1 in E1 and
      for e9 being object st e9 DJoins v,w,G & e9 in E1 holds e9 = e1
      by A17, Def6, GLIB_000:def 14;
    consider e2 being object such that
      e2 DJoins v,w,G & e2 in E2 and
      A19: for e9 being object st e9 DJoins v,w,G & e9 in E2 holds e9 = e2
      by A17, Def6, GLIB_000:def 14;
    consider v0,w0 being object such that
      A20: e1 DJoins v0,w0,G & f.e1 DJoins v0,w0,G by A10, A18;
    v0 = v & w0 = w by A18, A20, GLIB_000:125;
    then A21: f.e1 = e2 by A10, A18, A19, A20;
    y DJoins v,w,G by A17, GLIB_000:def 14;
    then y = e2 by A17, A19;
    hence y in rng f by A10, A18, A21, FUNCT_1:3;
  end;
  hence rng f = E2 by TARSKI:2;
  let e,v,w be object;
  assume e in E1;
  then consider v0,w0 being object such that
    A22: e DJoins v0,w0,G & f.e DJoins v0,w0,G by A10;
  hereby
    assume e DJoins v,w,G;
    then v0 = v & w0 = w by A22, GLIB_000:125;
    hence f.e DJoins v,w,G by A22;
  end;
  assume f.e DJoins v,w,G;
  then v0 = v & w0 = w by A22, GLIB_000:125;
  hence e DJoins v,w,G by A22;
end;
