Theorem resexd 6026

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 6025	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	syl 18	1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  Vcvv 3454   ↾ cres 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-in 3911  df-ss 3921  df-res 5672

This theorem is used by:  gsum2dlem2  20047  psrbagres  22091  tsmspropd  24300  ulmss  26571  elrgspnlem4  33574  extvfvcl  33935  esplyind  33974  lmimdim  34003  aks6d1c6lem3  42967  pwssplit4  43844  limsupresre  46438  limsupresico  46442  limsupresuz  46445  limsupres  46447  limsupresxr  46508  liminfresxr  46509  liminfresico  46513  liminfresre  46521  liminfresuz  46526  isubgriedg  48656  isubgrvtx  48660