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Theorem dmresi 6070
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 4008 . . 3 𝐴 ⊆ V
2 dmi 5932 . . 3 dom I = V
31, 2sseqtrri 4033 . 2 𝐴 ⊆ dom I
4 ssdmres 6031 . 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 1540  Vcvv 3480  wss 3951   I cid 5577  dom cdm 5685  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-dm 5695  df-res 5697
This theorem is referenced by:  iordsmo  8397  residfi  9378  hartogslem1  9582  dfac9  10177  hsmexlem5  10470  relexpdmg  15081  relexpfld  15088  relexpaddg  15092  dirdm  18645  islinds2  21833  lindsind2  21839  f1linds  21845  wilthlem3  27113  ausgrusgrb  29182  usgrres1  29332  usgrexilem  29457  filnetlem3  36381  filnetlem4  36382  rclexi  43628  dfrtrcl5  43642  dfrcl2  43687  brfvrcld2  43705  iunrelexp0  43715  relexpiidm  43717  relexp01min  43726  ushggricedg  47896  stgrusgra  47926  gpgusgra  48012  uspgrsprfo  48064
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