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Theorem opabresid 5327
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 5320 . 2  |-  ( {
<. x ,  y >.  |  y  =  x }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  x ) }
2 equcom 1743 . . . . 5  |-  ( y  =  x  <->  x  =  y )
32opabbii 4511 . . . 4  |-  { <. x ,  y >.  |  y  =  x }  =  { <. x ,  y
>.  |  x  =  y }
4 dfid3 4796 . . . 4  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
53, 4eqtr4i 2499 . . 3  |-  { <. x ,  y >.  |  y  =  x }  =  _I
65reseq1i 5269 . 2  |-  ( {
<. x ,  y >.  |  y  =  x }  |`  A )  =  (  _I  |`  A )
71, 6eqtr3i 2498 1  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   {copab 4504    _I cid 4790    |` cres 5001
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
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-ne 2664  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-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-res 5011
This theorem is referenced by:  mptresid  5328  pospo  15460  opsrtoslem1  17947  tgphaus  20378
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