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Theorem prtlem11 26734
Description: Lemma for prter2 26749. (Contributed by Rodolfo Medina, 12-Oct-2010.)
Assertion
Ref Expression
prtlem11  |-  ( B  e.  D  ->  ( C  e.  A  ->  ( B  =  [ C ]  .~  ->  B  e.  ( A /.  .~  )
) ) )

Proof of Theorem prtlem11
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 risset 2590 . . . 4  |-  ( C  e.  A  <->  E. x  e.  A  x  =  C )
2 r19.41v 2693 . . . . 5  |-  ( E. x  e.  A  ( x  =  C  /\  B  =  [ C ]  .~  )  <->  ( E. x  e.  A  x  =  C  /\  B  =  [ C ]  .~  ) )
3 eceq1 6696 . . . . . . 7  |-  ( x  =  C  ->  [ x ]  .~  =  [ C ]  .~  )
4 eqtr3 2302 . . . . . . . 8  |-  ( ( [ x ]  .~  =  [ C ]  .~  /\  B  =  [ C ]  .~  )  ->  [ x ]  .~  =  B )
54eqcomd 2288 . . . . . . 7  |-  ( ( [ x ]  .~  =  [ C ]  .~  /\  B  =  [ C ]  .~  )  ->  B  =  [ x ]  .~  )
63, 5sylan 457 . . . . . 6  |-  ( ( x  =  C  /\  B  =  [ C ]  .~  )  ->  B  =  [ x ]  .~  )
76reximi 2650 . . . . 5  |-  ( E. x  e.  A  ( x  =  C  /\  B  =  [ C ]  .~  )  ->  E. x  e.  A  B  =  [ x ]  .~  )
82, 7sylbir 204 . . . 4  |-  ( ( E. x  e.  A  x  =  C  /\  B  =  [ C ]  .~  )  ->  E. x  e.  A  B  =  [ x ]  .~  )
91, 8sylanb 458 . . 3  |-  ( ( C  e.  A  /\  B  =  [ C ]  .~  )  ->  E. x  e.  A  B  =  [ x ]  .~  )
10 elqsg 6711 . . 3  |-  ( B  e.  D  ->  ( B  e.  ( A /.  .~  )  <->  E. x  e.  A  B  =  [ x ]  .~  ) )
119, 10syl5ibr 212 . 2  |-  ( B  e.  D  ->  (
( C  e.  A  /\  B  =  [ C ]  .~  )  ->  B  e.  ( A /.  .~  ) ) )
1211exp3a 425 1  |-  ( B  e.  D  ->  ( C  e.  A  ->  ( B  =  [ C ]  .~  ->  B  e.  ( A /.  .~  )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   [cec 6658   /.cqs 6659
This theorem is referenced by:  prter2  26749
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-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-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662  df-qs 6666
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