Theorem resexg 5018

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 5002	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		ssexg 4199	. 2 ⊢ (((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	mpan 424	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2178  Vcvv 2776   ⊆ wss 3174   ↾ cres 4695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-sep 4178

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-res 4705

This theorem is referenced by:  resex  5019  offres  6243  resixp  6843  seqf1oglem2  10702  climres  11729  setsvalg  12977  setsex  12979  setsslid  12998  gsumsplit1r  13345  znval  14513  znle  14514  znbaslemnn  14516  znleval  14530