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Theorem dmresv 6219
Description: The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
Assertion
Ref Expression
dmresv dom (𝐴 ↾ V) = dom 𝐴

Proof of Theorem dmresv
StepHypRef Expression
1 dmres 6029 . 2 dom (𝐴 ↾ V) = (V ∩ dom 𝐴)
2 incom 4208 . 2 (V ∩ dom 𝐴) = (dom 𝐴 ∩ V)
3 inv1 4397 . 2 (dom 𝐴 ∩ V) = dom 𝐴
41, 2, 33eqtri 2768 1 dom (𝐴 ↾ V) = dom 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  Vcvv 3479  cin 3949  dom cdm 5684  cres 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-dm 5694  df-res 5696
This theorem is referenced by:  fidomdm  9375  dmttrcl  9762  dmct  10565
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