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Theorem dmresi 6051
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 4006 . . 3 𝐴 ⊆ V
2 dmi 5921 . . 3 dom I = V
31, 2sseqtrri 4019 . 2 𝐴 ⊆ dom I
4 ssdmres 6004 . 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 1541  Vcvv 3474  wss 3948   I cid 5573  dom cdm 5676  cres 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-dm 5686  df-res 5688
This theorem is referenced by:  iordsmo  8356  residfi  9332  hartogslem1  9536  dfac9  10130  hsmexlem5  10424  relexpdmg  14988  relexpfld  14995  relexpaddg  14999  dirdm  18552  islinds2  21367  lindsind2  21373  f1linds  21379  wilthlem3  26571  ausgrusgrb  28422  usgrres1  28569  usgrexilem  28694  filnetlem3  35260  filnetlem4  35261  rclexi  42356  dfrtrcl5  42370  dfrcl2  42415  brfvrcld2  42433  iunrelexp0  42443  relexpiidm  42445  relexp01min  42454  ushrisomgr  46499  uspgrsprfo  46516
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