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Theorem prtlem11 29152
Description: Lemma for prter2 29167. (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 2877 . . . 4  |-  ( C  e.  A  <->  E. x  e.  A  x  =  C )
2 r19.41v 2972 . . . . 5  |-  ( E. x  e.  A  ( x  =  C  /\  B  =  [ C ]  .~  )  <->  ( E. x  e.  A  x  =  C  /\  B  =  [ C ]  .~  ) )
3 eceq1 7240 . . . . . . 7  |-  ( x  =  C  ->  [ x ]  .~  =  [ C ]  .~  )
4 eqtr3 2479 . . . . . . . 8  |-  ( ( [ x ]  .~  =  [ C ]  .~  /\  B  =  [ C ]  .~  )  ->  [ x ]  .~  =  B )
54eqcomd 2459 . . . . . . 7  |-  ( ( [ x ]  .~  =  [ C ]  .~  /\  B  =  [ C ]  .~  )  ->  B  =  [ x ]  .~  )
63, 5sylan 471 . . . . . 6  |-  ( ( x  =  C  /\  B  =  [ C ]  .~  )  ->  B  =  [ x ]  .~  )
76reximi 2922 . . . . 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 7255 . . 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 1370    e. wcel 1758   E.wrex 2796   [cec 7202   /.cqs 7203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-cnv 4949  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-ec 7206  df-qs 7210
This theorem is referenced by:  prter2  29167
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