Theorem resexd 6015

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 6014	. 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 2145  Vcvv 3450   ↾ cres 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-in 3905  df-ss 3915  df-res 5659

This theorem is used by:  gsum2dlem2  20146  psrbagres  22199  tsmspropd  24412  ulmss  26687  elrgspnlem4  33739  extvfvcl  34101  esplyind  34140  lmimdim  34169  aks6d1c6lem3  43142  pwssplit4  44034  limsupresre  46628  limsupresico  46632  limsupresuz  46635  limsupres  46637  limsupresxr  46698  liminfresxr  46699  liminfresico  46703  liminfresre  46711  liminfresuz  46716  isubgriedg  48883  isubgrvtx  48887