MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptresid Unicode version

Theorem mptresid 5154
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 4228 . 2  |-  ( x  e.  A  |->  x )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }
2 opabresid 5153 . 2  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
31, 2eqtri 2424 1  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721   {copab 4225    e. cmpt 4226    _I cid 4453    |` cres 4839
This theorem is referenced by:  idref  5938  pwfseqlem5  8494  restid2  13613  curf2ndf  14299  hofcl  14311  yonedainv  14333  sylow1lem2  15188  sylow3lem1  15216  0frgp  15366  frgpcyg  16809  txswaphmeolem  17789  idnghm  18730  dvexp  19792  dvmptid  19796  mvth  19829  plyid  20081  coeidp  20134  dgrid  20135  plyremlem  20174  taylply2  20237  wilthlem2  20805  ftalem7  20814  zrhre  24338  qqhre  24339
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-res 4849
  Copyright terms: Public domain W3C validator