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Theorem prtlem11 31869
Description: Lemma for prter2 31884. (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 2931 . . . 4  |-  ( C  e.  A  <->  E. x  e.  A  x  =  C )
2 r19.41v 2958 . . . . 5  |-  ( E. x  e.  A  ( x  =  C  /\  B  =  [ C ]  .~  )  <->  ( E. x  e.  A  x  =  C  /\  B  =  [ C ]  .~  ) )
3 eceq1 7383 . . . . . . 7  |-  ( x  =  C  ->  [ x ]  .~  =  [ C ]  .~  )
4 eqtr3 2430 . . . . . . . 8  |-  ( ( [ x ]  .~  =  [ C ]  .~  /\  B  =  [ C ]  .~  )  ->  [ x ]  .~  =  B )
54eqcomd 2410 . . . . . . 7  |-  ( ( [ x ]  .~  =  [ C ]  .~  /\  B  =  [ C ]  .~  )  ->  B  =  [ x ]  .~  )
63, 5sylan 469 . . . . . 6  |-  ( ( x  =  C  /\  B  =  [ C ]  .~  )  ->  B  =  [ x ]  .~  )
76reximi 2871 . . . . 5  |-  ( E. x  e.  A  ( x  =  C  /\  B  =  [ C ]  .~  )  ->  E. x  e.  A  B  =  [ x ]  .~  )
82, 7sylbir 213 . . . 4  |-  ( ( E. x  e.  A  x  =  C  /\  B  =  [ C ]  .~  )  ->  E. x  e.  A  B  =  [ x ]  .~  )
91, 8sylanb 470 . . 3  |-  ( ( C  e.  A  /\  B  =  [ C ]  .~  )  ->  E. x  e.  A  B  =  [ x ]  .~  )
10 elqsg 7399 . . 3  |-  ( B  e.  D  ->  ( B  e.  ( A /.  .~  )  <->  E. x  e.  A  B  =  [ x ]  .~  ) )
119, 10syl5ibr 221 . 2  |-  ( B  e.  D  ->  (
( C  e.  A  /\  B  =  [ C ]  .~  )  ->  B  e.  ( A /.  .~  ) ) )
1211expd 434 1  |-  ( B  e.  D  ->  ( C  e.  A  ->  ( B  =  [ C ]  .~  ->  B  e.  ( A /.  .~  )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2754   [cec 7345   /.cqs 7346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ec 7349  df-qs 7353
This theorem is referenced by:  prter2  31884
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