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Theorem prtlem12 32150
Description: Lemma for prtex 32163 and prter3 32165. (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 4980 . 2  |-  Rel  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u
) }
2 releq 4937 . 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 236 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 370    = wceq 1437   E.wrex 2783   {copab 4483   Rel wrel 4859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-opab 4485  df-xp 4860  df-rel 4861
This theorem is referenced by: (None)
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