Theorem dmresv 6188

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

Step	Hyp	Ref	Expression
1		dmres 5999	. 2 ⊢ dom (𝐴 ↾ V) = (V ∩ dom 𝐴)
2		incom 4162	. 2 ⊢ (V ∩ dom 𝐴) = (dom 𝐴 ∩ V)
3		inv1 4353	. 2 ⊢ (dom 𝐴 ∩ V) = dom 𝐴
4	1, 2, 3	3eqtri 2790	1 ⊢ dom (𝐴 ↾ V) = dom 𝐴

Syntax hints:   = wceq 1561  Vcvv 3455   ∩ cin 3904  dom cdm 5648   ↾ cres 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-dm 5658  df-res 5660

This theorem is referenced by:  fidomdm  9278  dmttrcl  9677  dmct  10482  dfsucmap3  38963