Theorem resexd 5991

Description: The restriction of a set is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
resexd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
resexd	⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ V)

Proof of Theorem resexd

Step	Hyp	Ref	Expression
1		resexd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		resexg 5990	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3430   ↾ cres 5630

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-in 3897  df-ss 3907  df-res 5640

This theorem is referenced by:  gsum2dlem2  19943  tsmspropd  24094  ulmss  26359  elrgspnlem4  33303  extvfvcl  33677  esplyind  33716  lmimdim  33745  aks6d1c6lem3  42608  psrbagres  42986  pwssplit4  43514  limsupresre  46121  limsupresico  46125  limsupresuz  46128  limsupres  46130  limsupresxr  46191  liminfresxr  46192  liminfresico  46196  liminfresre  46204  liminfresuz  46209  isubgriedg  48330  isubgrvtx  48334