Theorem resexg 5981

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 5955	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		ssexg 5253	. 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 3427   ⊆ wss 3885   ↾ cres 5622

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 2707  ax-sep 5220

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 2714  df-cleq 2727  df-clel 2810  df-rab 3388  df-v 3429  df-in 3892  df-ss 3902  df-res 5632

This theorem is referenced by:  resexd  5982  resex  5983  fvtresfn  6939  offres  7925  ressuppss  8122  ressuppssdif  8124  ecelqsw  8704  uniqsw  8710  eceldmqs  8723  resixp  8870  f1imaen3g  8952  dif1enlem  9083  sbthfilem  9121  fsuppres  9295  climres  15526  setsvalg  17125  setsid  17166  symgfixels  19398  qtopres  23651  vtxdginducedm1  29600  redwlk  29727  hhssva  31316  hhsssm  31317  hhshsslem1  31326  resf1o  32791  eulerpartlemmf  34507  exidres  38187  exidresid  38188  xrnresex  38738  unidmqs  39048  disjqmap2  39135  lmhmlnmsplit  43503  climresdm  46266  setsv  47826