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Theorem dmresi 6009
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 3972 . . 3 𝐴 ⊆ V
2 dmi 5881 . . 3 dom I = V
31, 2sseqtrri 3985 . 2 𝐴 ⊆ dom I
4 ssdmres 5964 . 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 3447  wss 3914   I cid 5534  dom cdm 5637  cres 5639
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 5260  ax-nul 5267  ax-pr 5388
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 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-dm 5647  df-res 5649
This theorem is referenced by:  iordsmo  8307  residfi  9283  hartogslem1  9486  dfac9  10080  hsmexlem5  10374  relexpdmg  14936  relexpfld  14943  relexpaddg  14947  dirdm  18497  islinds2  21242  lindsind2  21248  f1linds  21254  wilthlem3  26442  ausgrusgrb  28165  usgrres1  28312  usgrexilem  28437  filnetlem3  34905  filnetlem4  34906  rclexi  41979  dfrtrcl5  41993  dfrcl2  42038  brfvrcld2  42056  iunrelexp0  42066  relexpiidm  42068  relexp01min  42077  ushrisomgr  46123  uspgrsprfo  46140
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