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Theorem prtlem12 26718
Description: Lemma for prtex 26731 and prter3 26733. (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 5003 . 2  |-  Rel  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u
) }
2 releq 4961 . 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 226 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 360    = wceq 1653   E.wrex 2708   {copab 4267   Rel wrel 4885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886  df-rel 4887
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