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Theorem mptresid 5318
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 4497 . 2  |-  ( x  e.  A  |->  x )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }
2 opabresid 5317 . 2  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
31, 2eqtri 2472 1  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1383    e. wcel 1804   {copab 4494    |-> cmpt 4495    _I cid 4780    |` cres 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-res 5001
This theorem is referenced by:  idref  6138  2fvcoidd  6185  pwfseqlem5  9044  restid2  14705  curf2ndf  15390  hofcl  15402  yonedainv  15424  sylow1lem2  16493  sylow3lem1  16521  0frgp  16671  frgpcyg  18485  evpmodpmf1o  18505  txswaphmeolem  20178  idnghm  21123  dvexp  22229  dvmptid  22233  mvth  22266  plyid  22479  coeidp  22532  dgrid  22533  plyremlem  22572  taylply2  22635  wilthlem2  23215  ftalem7  23224  zrhre  27870  qqhre  27871  fourierdlem60  31838  fourierdlem61  31839  usgfis  32284  usgfisALT  32288
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