Theorem dmresexg 6013

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 6011	. 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2		inex1g 5288	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∩ dom 𝐴) ∈ V)
3	1, 2	eqeltrid 2867	1 ⊢ (𝐵 ∈ 𝑉 → dom (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2143  Vcvv 3455   ∩ cin 3904  dom cdm 5661   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-dm 5671  df-res 5673

This theorem is referenced by:  resfunexg  7213  resfunexgALT  7941