Theorem dmresv 6194

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 6004	. 2 ⊢ dom (𝐴 ↾ V) = (V ∩ dom 𝐴)
2		incom 4189	. 2 ⊢ (V ∩ dom 𝐴) = (dom 𝐴 ∩ V)
3		inv1 4378	. 2 ⊢ (dom 𝐴 ∩ V) = dom 𝐴
4	1, 2, 3	3eqtri 2763	1 ⊢ dom (𝐴 ↾ V) = dom 𝐴

Syntax hints:   = wceq 1540  Vcvv 3464   ∩ cin 3930  dom cdm 5659   ↾ cres 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-dm 5669  df-res 5671

This theorem is referenced by:  fidomdm  9351  dmttrcl  9740  dmct  10543