Theorem resexg 6025

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 6004	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		ssexg 5317	. 2 ⊢ (((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	mpan 689	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2099  Vcvv 3470   ⊆ wss 3945   ↾ cres 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-in 3952  df-ss 3962  df-res 5684

This theorem is referenced by:  resexd  6026  resex  6027  fvtresfn  7001  offres  7981  ressuppss  8181  ressuppssdif  8183  resixp  8945  dif1enlem  9174  dif1enlemOLD  9175  sbthfilem  9219  fsuppres  9410  climres  15545  setsvalg  17128  setsid  17170  symgfixels  19382  qtopres  23595  vtxdginducedm1  29350  redwlk  29479  hhssva  31060  hhsssm  31061  hhshsslem1  31070  resf1o  32506  eulerpartlemmf  33989  exidres  37345  exidresid  37346  xrnresex  37872  unidmqs  38120  lmhmlnmsplit  42505  climresdm  45232  setsv  46712