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Theorem prtlem12 26838
Description: Lemma for prtex 26851 and prter3 26853. (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 4828 . 2  |-  Rel  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u
) }
2 releq 4787 . 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 1632    e. wcel 1696   E.wrex 2557   {copab 4092   Rel wrel 4710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-rel 4712
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