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Theorem dmresi 5005
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 3198 . . 3  |-  A  C_  _V
2 dmi 4893 . . 3  |-  dom  _I  =  _V
31, 2sseqtr4i 3211 . 2  |-  A  C_  dom  _I
4 ssdmres 4977 . 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 1623   _Vcvv 2788    C_ wss 3152    _I cid 4304   dom cdm 4689    |` cres 4691
This theorem is referenced by:  fnresi  5361  iordsmo  6374  hartogslem1  7257  dfac9  7762  hsmexlem5  8056  dirdm  14356  wilthlem2  20307  wilthlem3  20308  relexpdm  24032  dispos  25287  filnetlem3  26329  filnetlem4  26330  islinds2  27283  lindsind2  27289  f1linds  27295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-dm 4699  df-res 4701
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