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Theorem opabresid 5196
Description: The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
Assertion
Ref Expression
opabresid  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
Distinct variable group:    x, A, y

Proof of Theorem opabresid
StepHypRef Expression
1 resopab 5189 . 2  |-  ( {
<. x ,  y >.  |  y  =  x }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  x ) }
2 equcom 1693 . . . . 5  |-  ( y  =  x  <->  x  =  y )
32opabbii 4274 . . . 4  |-  { <. x ,  y >.  |  y  =  x }  =  { <. x ,  y
>.  |  x  =  y }
4 dfid3 4501 . . . 4  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
53, 4eqtr4i 2461 . . 3  |-  { <. x ,  y >.  |  y  =  x }  =  _I
65reseq1i 5144 . 2  |-  ( {
<. x ,  y >.  |  y  =  x }  |`  A )  =  (  _I  |`  A )
71, 6eqtr3i 2460 1  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   {copab 4267    _I cid 4495    |` cres 4882
This theorem is referenced by:  mptresid  5197  pospo  14432  opsrtoslem1  16546  tgphaus  18148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-res 4892
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