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Theorem prtlem12 30536
Description: Lemma for prtex 30549 and prter3 30551. (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 5135 . 2  |-  Rel  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u
) }
2 releq 5091 . 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 233 1  |-  (  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }  ->  Rel 
.~  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wrex 2818   {copab 4510   Rel wrel 5010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-opab 4512  df-xp 5011  df-rel 5012
This theorem is referenced by: (None)
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