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Theorem mptresid 5135
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 4209 . 2  |-  ( x  e.  A  |->  x )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }
2 opabresid 5134 . 2  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
31, 2eqtri 2407 1  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717   {copab 4206    e. cmpt 4207    _I cid 4434    |` cres 4820
This theorem is referenced by:  idref  5918  pwfseqlem5  8471  restid2  13585  curf2ndf  14271  hofcl  14283  yonedainv  14305  sylow1lem2  15160  sylow3lem1  15188  0frgp  15338  frgpcyg  16777  txswaphmeolem  17757  idnghm  18648  dvexp  19706  dvmptid  19710  mvth  19743  plyid  19995  coeidp  20048  dgrid  20049  plyremlem  20088  taylply2  20151  wilthlem2  20719  ftalem7  20728  zrhre  24181  qqhre  24182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-res 4830
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