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Theorem dmresv 5555
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 5382 . 2 dom (𝐴 ↾ V) = (V ∩ dom 𝐴)
2 incom 3788 . 2 (V ∩ dom 𝐴) = (dom 𝐴 ∩ V)
3 inv1 3947 . 2 (dom 𝐴 ∩ V) = dom 𝐴
41, 2, 33eqtri 2652 1 dom (𝐴 ↾ V) = dom 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  Vcvv 3191  cin 3559  dom cdm 5079  cres 5081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-dm 5089  df-res 5091
This theorem is referenced by:  fidomdm  8188  dmct  9291
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