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Theorem prtlem12 26735
Description: Lemma for prtex 26748 and prter3 26750. (Contributed by Rodolfo Medina, 13-Oct-2010.)
Assertion
Ref Expression
prtlem12  |-  (  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }  ->  Rel 
.~  )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y, u)    .~ ( x, y, u)

Proof of Theorem prtlem12
StepHypRef Expression
1 relopab 4812 . 2  |-  Rel  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u
) }
2 releq 4771 . 2  |-  (  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }  ->  ( Rel  .~  <->  Rel  { <. x ,  y >.  |  E. u  e.  A  (
x  e.  u  /\  y  e.  u ) } ) )
31, 2mpbiri 224 1  |-  (  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }  ->  Rel 
.~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {copab 4076   Rel wrel 4694
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-ral 2548  df-rex 2549  df-rab 2552  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
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