Theorem resexd 5985

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 5984	. 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 5624

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 5231

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 5634

This theorem is referenced by:  gsum2dlem2  19904  tsmspropd  24075  ulmss  26346  elrgspnlem4  33311  extvfvcl  33685  esplyind  33724  lmimdim  33753  aks6d1c6lem3  42603  psrbagres  42988  pwssplit4  43520  limsupresre  46128  limsupresico  46132  limsupresuz  46135  limsupres  46137  limsupresxr  46198  liminfresxr  46199  liminfresico  46203  liminfresre  46211  liminfresuz  46216  isubgriedg  48297  isubgrvtx  48301