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 5999	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		ssexg 5289	. 2 ⊢ (((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	mpan 702	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904   ↾ cres 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-in 3911  df-ss 3921  df-res 5672

This theorem is used by:  resexd  6026  resex  6027  fvtresfn  6992  offres  7978  ressuppss  8177  ressuppssdif  8179  ecelqsw  8764  uniqsw  8770  eceldmqs  8783  resixp  8929  f1imaen3g  9011  dif1enlem  9142  sbthfilem  9180  fsuppres  9351  climres  15633  setsvalg  17232  setsid  17273  symgfixels  19510  qtopres  23866  vtxdginducedm1  29904  redwlk  30031  hhssva  31620  hhsssm  31621  hhshsslem1  31630  resf1o  33086  eulerpartlemmf  34774  exidres  38557  exidresid  38558  xrnresex  39106  unidmqs  39416  disjqmap2  39503  lmhmlnmsplit  43842  climresdm  46592  setsv  48155