Theorem resexg 5477

Description: The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
resexg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)

Proof of Theorem resexg

Step	Hyp	Ref	Expression
1		resss 5457	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		ssexg 4837	. 2 ⊢ (((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	mpan 706	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607   ↾ cres 5145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-res 5155

This theorem is referenced by:  resex  5478  fvtresfn  6323  offres  7205  ressuppss  7359  ressuppssdif  7361  resixp  7985  fsuppres  8341  climres  14350  setsvalg  15934  setsid  15961  symgfixels  17900  gsumval3lem1  18352  gsumval3lem2  18353  gsum2dlem2  18416  qtopres  21549  tsmspropd  21982  ulmss  24196  vtxdginducedm1  26495  redwlk  26625  hhssva  28242  hhsssm  28243  hhshsslem1  28252  resf1o  29633  eulerpartlemmf  30565  exidres  33807  exidresid  33808  xrnresex  34304  lmhmlnmsplit  37974  pwssplit4  37976  resexd  39635  setsv  41673