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Theorem dmresi 6058
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 4004 . . 3 𝐴 ⊆ V
2 dmi 5926 . . 3 dom I = V
31, 2sseqtrri 4017 . 2 𝐴 ⊆ dom I
4 ssdmres 6020 . 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 1533  Vcvv 3471  wss 3947   I cid 5577  dom cdm 5680  cres 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-id 5578  df-xp 5686  df-rel 5687  df-dm 5690  df-res 5692
This theorem is referenced by:  iordsmo  8382  residfi  9363  hartogslem1  9571  dfac9  10165  hsmexlem5  10459  relexpdmg  15027  relexpfld  15034  relexpaddg  15038  dirdm  18597  islinds2  21752  lindsind2  21758  f1linds  21764  wilthlem3  27020  ausgrusgrb  28996  usgrres1  29146  usgrexilem  29271  filnetlem3  35869  filnetlem4  35870  rclexi  43048  dfrtrcl5  43062  dfrcl2  43107  brfvrcld2  43125  iunrelexp0  43135  relexpiidm  43137  relexp01min  43146  ushggricedg  47244  uspgrsprfo  47261
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