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Theorem mptresid 5020
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 4095 . 2  |-  ( x  e.  A  |->  x )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }
2 opabresid 5019 . 2  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
31, 2eqtri 2316 1  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   {copab 4092    e. cmpt 4093    _I cid 4320    |` cres 4707
This theorem is referenced by:  idref  5775  pwfseqlem5  8301  restid2  13351  curf2ndf  14037  hofcl  14049  yonedainv  14071  sylow1lem2  14926  sylow3lem1  14954  0frgp  15104  frgpcyg  16543  txswaphmeolem  17511  idnghm  18268  dvexp  19318  dvmptid  19322  mvth  19355  plyid  19607  coeidp  19660  dgrid  19661  plyremlem  19700  taylply2  19763  wilthlem2  20323  ftalem7  20332  crimmt1  25249  crimmt2  25250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-res 4717
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