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

Step	Hyp	Ref	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
4	1, 2, 3	3eqtri 2768	1 ⊢ dom (𝐴 ↾ V) = dom 𝐴

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