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Theorem opabresid 4960
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 4951 . 2  |-  ( {
<. x ,  y >.  |  y  =  x }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  x ) }
2 equcom 1706 . . . . 5  |-  ( y  =  x  <->  x  =  y )
32opabbii 4070 . . . 4  |-  { <. x ,  y >.  |  y  =  x }  =  { <. x ,  y
>.  |  x  =  y }
4 df-id 4293 . . . 4  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
53, 4eqtr4i 2201 . . 3  |-  { <. x ,  y >.  |  y  =  x }  =  _I
65reseq1i 4903 . 2  |-  ( {
<. x ,  y >.  |  y  =  x }  |`  A )  =  (  _I  |`  A )
71, 6eqtr3i 2200 1  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353    e. wcel 2148   {copab 4063    _I cid 4288    |` cres 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-res 4638
This theorem is referenced by:  mptresid  4961
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