Theorem resexd 5988

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 5987	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3444   ↾ cres 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-in 3918  df-ss 3928  df-res 5643

This theorem is referenced by:  gsum2dlem2  19886  tsmspropd  24053  ulmss  26340  elrgspnlem4  33213  lmimdim  33593  aks6d1c6lem3  42154  psrbagres  42528  pwssplit4  43072  limsupresre  45688  limsupresico  45692  limsupresuz  45695  limsupres  45697  limsupresxr  45758  liminfresxr  45759  liminfresico  45763  liminfresre  45771  liminfresuz  45776  isubgriedg  47857  isubgrvtx  47861