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Theorem prtlem11 30211
Description: Lemma for prter2 30226. (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 2987 . . . 4  |-  ( C  e.  A  <->  E. x  e.  A  x  =  C )
2 r19.41v 3014 . . . . 5  |-  ( E. x  e.  A  ( x  =  C  /\  B  =  [ C ]  .~  )  <->  ( E. x  e.  A  x  =  C  /\  B  =  [ C ]  .~  ) )
3 eceq1 7344 . . . . . . 7  |-  ( x  =  C  ->  [ x ]  .~  =  [ C ]  .~  )
4 eqtr3 2495 . . . . . . . 8  |-  ( ( [ x ]  .~  =  [ C ]  .~  /\  B  =  [ C ]  .~  )  ->  [ x ]  .~  =  B )
54eqcomd 2475 . . . . . . 7  |-  ( ( [ x ]  .~  =  [ C ]  .~  /\  B  =  [ C ]  .~  )  ->  B  =  [ x ]  .~  )
63, 5sylan 471 . . . . . 6  |-  ( ( x  =  C  /\  B  =  [ C ]  .~  )  ->  B  =  [ x ]  .~  )
76reximi 2932 . . . . 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 472 . . 3  |-  ( ( C  e.  A  /\  B  =  [ C ]  .~  )  ->  E. x  e.  A  B  =  [ x ]  .~  )
10 elqsg 7360 . . 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 436 1  |-  ( B  e.  D  ->  ( C  e.  A  ->  ( B  =  [ C ]  .~  ->  B  e.  ( A /.  .~  )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   [cec 7306   /.cqs 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-ec 7310  df-qs 7314
This theorem is referenced by:  prter2  30226
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