Theorem dmresv 6231

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 6041	. 2 ⊢ dom (𝐴 ↾ V) = (V ∩ dom 𝐴)
2		incom 4230	. 2 ⊢ (V ∩ dom 𝐴) = (dom 𝐴 ∩ V)
3		inv1 4421	. 2 ⊢ (dom 𝐴 ∩ V) = dom 𝐴
4	1, 2, 3	3eqtri 2772	1 ⊢ dom (𝐴 ↾ V) = dom 𝐴

Syntax hints:   = wceq 1537  Vcvv 3488   ∩ cin 3975  dom cdm 5700   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-dm 5710  df-res 5712

This theorem is referenced by:  fidomdm  9402  dmttrcl  9790  dmct  10593