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Theorem dmresi 5021
Description: The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
Assertion
Ref Expression
dmresi  |-  dom  (  _I  |`  A )  =  A

Proof of Theorem dmresi
StepHypRef Expression
1 ssv 3211 . . 3  |-  A  C_  _V
2 dmi 4909 . . 3  |-  dom  _I  =  _V
31, 2sseqtr4i 3224 . 2  |-  A  C_  dom  _I
4 ssdmres 4993 . 2  |-  ( A 
C_  dom  _I  <->  dom  (  _I  |`  A )  =  A )
53, 4mpbi 199 1  |-  dom  (  _I  |`  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801    C_ wss 3165    _I cid 4320   dom cdm 4705    |` cres 4707
This theorem is referenced by:  fnresi  5377  iordsmo  6390  hartogslem1  7273  dfac9  7778  hsmexlem5  8072  dirdm  14372  wilthlem2  20323  wilthlem3  20324  relexpdm  24047  dispos  25390  filnetlem3  26432  filnetlem4  26433  islinds2  27386  lindsind2  27392  f1linds  27398
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-eu 2160  df-mo 2161  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-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-dm 4715  df-res 4717
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