Theorem resexd 5986

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 5985	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3439   ↾ cres 5625

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 2707  ax-sep 5240

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 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-in 3907  df-ss 3917  df-res 5635

This theorem is referenced by:  gsum2dlem2  19902  tsmspropd  24078  ulmss  26364  elrgspnlem4  33306  extvfvcl  33680  esplyind  33710  lmimdim  33739  aks6d1c6lem3  42461  psrbagres  42836  pwssplit4  43368  limsupresre  45977  limsupresico  45981  limsupresuz  45984  limsupres  45986  limsupresxr  46047  liminfresxr  46048  liminfresico  46052  liminfresre  46060  liminfresuz  46065  isubgriedg  48146  isubgrvtx  48150