MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqrelrdv2 Unicode version

Theorem eqrelrdv2 4786
Description: A version of eqrelrdv 4783. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Hypothesis
Ref Expression
eqrelrdv2.1  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
Assertion
Ref Expression
eqrelrdv2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y

Proof of Theorem eqrelrdv2
StepHypRef Expression
1 eqrelrdv2.1 . . . 4  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
21alrimivv 1618 . . 3  |-  ( ph  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
32adantl 452 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
4 eqrel 4777 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
54adantr 451 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
63, 5mpbird 223 1  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   <.cop 3643   Rel wrel 4694
This theorem is referenced by:  xpiindi  4821  fliftcnv  5810  dmtpos  6246  ercnv  6681  fpwwe2lem9  8260  invsym2  13665  eqbrrdv2  26143  dibglbN  30729  diclspsn  30757  dih1  30849  dihglbcpreN  30863  dihmeetlem4preN  30869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696
  Copyright terms: Public domain W3C validator