Theorem resexd 6028

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 6027	. 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 2147  Vcvv 3454   ↾ cres 5657

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-res 5667

This theorem is used by:  gsum2dlem2  20212  psrbagres  22265  tsmspropd  24478  ulmss  26751  elrgspnlem4  33873  extvfvcl  34235  esplyind  34274  lmimdim  34303  gdlop4fv  35965  aks6d1c6lem3  43413  pwssplit4  44282  limsupresre  46909  limsupresico  46913  limsupresuz  46916  limsupres  46918  limsupresxr  46979  liminfresxr  46980  liminfresico  46984  liminfresre  46992  liminfresuz  46997  isubgriedg  49164  isubgrvtx  49168