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Theorem dmresi 6071
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 4019 . . 3 𝐴 ⊆ V
2 dmi 5934 . . 3 dom I = V
31, 2sseqtrri 4032 . 2 𝐴 ⊆ dom I
4 ssdmres 6032 . 2 (𝐴 ⊆ dom I ↔ dom ( I ↾ 𝐴) = 𝐴)
53, 4mpbi 230 1 dom ( I ↾ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  Vcvv 3477  wss 3962   I cid 5581  dom cdm 5688  cres 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-dm 5698  df-res 5700
This theorem is referenced by:  iordsmo  8395  residfi  9375  hartogslem1  9579  dfac9  10174  hsmexlem5  10467  relexpdmg  15077  relexpfld  15084  relexpaddg  15088  dirdm  18657  islinds2  21850  lindsind2  21856  f1linds  21862  wilthlem3  27127  ausgrusgrb  29196  usgrres1  29346  usgrexilem  29471  filnetlem3  36362  filnetlem4  36363  rclexi  43604  dfrtrcl5  43618  dfrcl2  43663  brfvrcld2  43681  iunrelexp0  43691  relexpiidm  43693  relexp01min  43702  ushggricedg  47833  stgrusgra  47861  gpgusgra  47946  uspgrsprfo  47991
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