Theorem resexg 6044

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 6018	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		ssexg 5322	. 2 ⊢ (((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	mpan 690	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3479   ⊆ wss 3950   ↾ cres 5686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-in 3957  df-ss 3967  df-res 5696

This theorem is referenced by:  resexd  6045  resex  6046  fvtresfn  7017  offres  8009  ressuppss  8209  ressuppssdif  8211  resixp  8974  f1imaen3g  9057  dif1enlem  9197  dif1enlemOLD  9198  sbthfilem  9239  fsuppres  9434  climres  15612  setsvalg  17204  setsid  17245  symgfixels  19453  qtopres  23707  vtxdginducedm1  29562  redwlk  29691  hhssva  31277  hhsssm  31278  hhshsslem1  31287  resf1o  32742  eulerpartlemmf  34378  exidres  37886  exidresid  37887  xrnresex  38408  unidmqs  38656  lmhmlnmsplit  43104  climresdm  45870  setsv  47370