Theorem resexg 6024

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 5998	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		ssexg 5288	. 2 ⊢ (((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	mpan 703	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902   ↾ cres 5661

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-in 3909  df-ss 3919  df-res 5671

This theorem is used by:  resexd  6025  resex  6026  fvtresfn  6993  offres  7983  ressuppss  8184  ressuppssdif  8186  ecelqsw  8771  uniqsw  8777  eceldmqs  8790  resixp  8943  f1imaen3g  9025  dif1enlem  9157  sbthfilem  9195  fsuppres  9366  climres  15664  setsvalg  17262  setsid  17303  symgfixels  19562  qtopres  23925  vtxdginducedm1  29989  redwlk  30116  hhssva  31724  hhsssm  31725  hhshsslem1  31734  resf1o  33188  eulerpartlemmf  34873  exidres  38615  exidresid  38616  xrnresex  39164  unidmqs  39474  disjqmap2  39561  lmhmlnmsplit  43915  climresdm  46665  setsv  48265