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Theorem mptresid 4660
Description: The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
Assertion
Ref Expression
mptresid  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
Distinct variable group:    x, A

Proof of Theorem mptresid
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-mpt 3820 . 2  |-  ( x  e.  A  |->  x )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }
2 opabresid 4659 . 2  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
31, 2eqtri 2060 1  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    = wceq 1243    e. wcel 1393   {copab 3817    |-> cmpt 3818    _I cid 4025    |` cres 4347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-res 4357
This theorem is referenced by:  idref  5396
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