Theorem dmresexg 5852

Description: The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)

Assertion

Ref	Expression
dmresexg	⊢ (𝐵 ∈ 𝑉 → dom (𝐴 ↾ 𝐵) ∈ V)

Proof of Theorem dmresexg

Step	Hyp	Ref	Expression
1		dmres 5850	. 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2		inex1g 5193	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∩ dom 𝐴) ∈ V)
3	1, 2	eqeltrid 2856	1 ⊢ (𝐵 ∈ 𝑉 → dom (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3409   ∩ cin 3859  dom cdm 5528   ↾ cres 5530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-xp 5534  df-dm 5538  df-res 5540

This theorem is referenced by:  resfunexg  6975  resfunexgALT  7659