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Theorem mptresid 5004
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 4079 . 2  |-  ( x  e.  A  |->  x )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }
2 opabresid 5003 . 2  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
31, 2eqtri 2303 1  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {copab 4076    e. cmpt 4077    _I cid 4304    |` cres 4691
This theorem is referenced by:  idref  5759  pwfseqlem5  8285  restid2  13335  curf2ndf  14021  hofcl  14033  yonedainv  14055  sylow1lem2  14910  sylow3lem1  14938  0frgp  15088  frgpcyg  16527  txswaphmeolem  17495  idnghm  18252  dvexp  19302  dvmptid  19306  mvth  19339  plyid  19591  coeidp  19644  dgrid  19645  plyremlem  19684  taylply2  19747  wilthlem2  20307  ftalem7  20316  crimmt1  25146  crimmt2  25147
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-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-res 4701
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