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Theorem dmresi 5196
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 3368 . . 3  |-  A  C_  _V
2 dmi 5084 . . 3  |-  dom  _I  =  _V
31, 2sseqtr4i 3381 . 2  |-  A  C_  dom  _I
4 ssdmres 5168 . 2  |-  ( A 
C_  dom  _I  <->  dom  (  _I  |`  A )  =  A )
53, 4mpbi 200 1  |-  dom  (  _I  |`  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1652   _Vcvv 2956    C_ wss 3320    _I cid 4493   dom cdm 4878    |` cres 4880
This theorem is referenced by:  fnresi  5562  iordsmo  6619  hartogslem1  7511  dfac9  8016  hsmexlem5  8310  dirdm  14679  wilthlem2  20852  wilthlem3  20853  ausisusgra  21380  cusgraexilem2  21476  relexpdm  25135  filnetlem3  26409  filnetlem4  26410  islinds2  27260  lindsind2  27266  f1linds  27272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-dm 4888  df-res 4890
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