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Theorem mptresid 5157
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 4349 . 2  |-  ( x  e.  A  |->  x )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }
2 opabresid 5156 . 2  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
31, 2eqtri 2461 1  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1364    e. wcel 1761   {copab 4346    e. cmpt 4347    _I cid 4627    |` cres 4838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-res 4848
This theorem is referenced by:  idref  5955  pwfseqlem5  8826  restid2  14365  curf2ndf  15053  hofcl  15065  yonedainv  15087  sylow1lem2  16091  sylow3lem1  16119  0frgp  16269  frgpcyg  17965  evpmodpmf1o  17985  txswaphmeolem  19336  idnghm  20281  dvexp  21386  dvmptid  21390  mvth  21423  plyid  21636  coeidp  21689  dgrid  21690  plyremlem  21729  taylply2  21792  wilthlem2  22366  ftalem7  22375  zrhre  26381  qqhre  26382
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