Theorem resexg 5949

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 5928	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		ssexg 5256	. 2 ⊢ (((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	mpan 688	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2104  Vcvv 3437   ⊆ wss 3892   ↾ cres 5602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3287  df-v 3439  df-in 3899  df-ss 3909  df-res 5612

This theorem is referenced by:  resexd  5950  resex  5951  fvtresfn  6909  offres  7858  ressuppss  8030  ressuppssdif  8032  resixp  8752  dif1enlem  8981  sbthfilem  9022  fsuppres  9197  climres  15329  setsvalg  16912  setsid  16954  symgfixels  19087  qtopres  22894  vtxdginducedm1  27955  redwlk  28085  hhssva  29664  hhsssm  29665  hhshsslem1  29674  resf1o  31110  eulerpartlemmf  32387  exidres  36080  exidresid  36081  xrnresex  36574  unidmqs  36808  lmhmlnmsplit  40950  climresdm  43440  setsv  44888