Theorem dmresv 6198

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 6001	. 2 ⊢ dom (𝐴 ↾ V) = (V ∩ dom 𝐴)
2		incom 4197	. 2 ⊢ (V ∩ dom 𝐴) = (dom 𝐴 ∩ V)
3		inv1 4390	. 2 ⊢ (dom 𝐴 ∩ V) = dom 𝐴
4	1, 2, 3	3eqtri 2759	1 ⊢ dom (𝐴 ↾ V) = dom 𝐴

Syntax hints:   = wceq 1534  Vcvv 3469   ∩ cin 3943  dom cdm 5672   ↾ cres 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-dm 5682  df-res 5684

This theorem is referenced by:  fidomdm  9347  dmttrcl  9738  dmct  10541