Theorem resexg 5990

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 5964	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		ssexg 5263	. 2 ⊢ (((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	mpan 691	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890   ↾ cres 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-in 3897  df-ss 3907  df-res 5640

This theorem is referenced by:  resexd  5991  resex  5992  fvtresfn  6948  offres  7933  ressuppss  8130  ressuppssdif  8132  ecelqsw  8712  uniqsw  8718  eceldmqs  8731  resixp  8878  f1imaen3g  8960  dif1enlem  9091  sbthfilem  9129  fsuppres  9303  climres  15534  setsvalg  17133  setsid  17174  symgfixels  19406  qtopres  23679  vtxdginducedm1  29633  redwlk  29760  hhssva  31349  hhsssm  31350  hhshsslem1  31359  resf1o  32824  eulerpartlemmf  34541  exidres  38221  exidresid  38222  xrnresex  38772  unidmqs  39082  disjqmap2  39169  lmhmlnmsplit  43541  climresdm  46304  setsv  47858