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Theorem dmresi 6052
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 4007 . . 3 𝐴 ⊆ V
2 dmi 5922 . . 3 dom I = V
31, 2sseqtrri 4020 . 2 𝐴 ⊆ dom I
4 ssdmres 6005 . 2 (𝐴 ⊆ dom I ↔ dom ( I ↾ 𝐴) = 𝐴)
53, 4mpbi 229 1 dom ( I ↾ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3475  wss 3949   I cid 5574  dom cdm 5677  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-dm 5687  df-res 5689
This theorem is referenced by:  iordsmo  8357  residfi  9333  hartogslem1  9537  dfac9  10131  hsmexlem5  10425  relexpdmg  14989  relexpfld  14996  relexpaddg  15000  dirdm  18553  islinds2  21368  lindsind2  21374  f1linds  21380  wilthlem3  26574  ausgrusgrb  28456  usgrres1  28603  usgrexilem  28728  filnetlem3  35313  filnetlem4  35314  rclexi  42414  dfrtrcl5  42428  dfrcl2  42473  brfvrcld2  42491  iunrelexp0  42501  relexpiidm  42503  relexp01min  42512  ushrisomgr  46557  uspgrsprfo  46574
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