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Theorem dmresi 5155
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 3328 . . 3  |-  A  C_  _V
2 dmi 5043 . . 3  |-  dom  _I  =  _V
31, 2sseqtr4i 3341 . 2  |-  A  C_  dom  _I
4 ssdmres 5127 . 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 1649   _Vcvv 2916    C_ wss 3280    _I cid 4453   dom cdm 4837    |` cres 4839
This theorem is referenced by:  fnresi  5521  iordsmo  6578  hartogslem1  7467  dfac9  7972  hsmexlem5  8266  dirdm  14634  wilthlem2  20805  wilthlem3  20806  ausisusgra  21333  cusgraexilem2  21429  relexpdm  25088  filnetlem3  26299  filnetlem4  26300  islinds2  27151  lindsind2  27157  f1linds  27163
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-eu 2258  df-mo 2259  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-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-dm 4847  df-res 4849
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