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Theorem opabresid 5003
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 4996 . 2  |-  ( {
<. x ,  y >.  |  y  =  x }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  x ) }
2 equcom 1647 . . . . 5  |-  ( y  =  x  <->  x  =  y )
32opabbii 4083 . . . 4  |-  { <. x ,  y >.  |  y  =  x }  =  { <. x ,  y
>.  |  x  =  y }
4 dfid3 4310 . . . 4  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
53, 4eqtr4i 2306 . . 3  |-  { <. x ,  y >.  |  y  =  x }  =  _I
65reseq1i 4951 . 2  |-  ( {
<. x ,  y >.  |  y  =  x }  |`  A )  =  (  _I  |`  A )
71, 6eqtr3i 2305 1  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {copab 4076    _I cid 4304    |` cres 4691
This theorem is referenced by:  mptresid  5004  pospo  14107  opsrtoslem1  16225  tgphaus  17799
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-ne 2448  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-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-res 4701
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