Theorem resex 5060

Description: The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)

Hypothesis

Ref	Expression
resex.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
resex	⊢ (𝐴 ↾ 𝐵) ∈ V

Proof of Theorem resex

Step	Hyp	Ref	Expression
1		resex.1	. 2 ⊢ 𝐴 ∈ V
2		resexg 5059	. 2 ⊢ (𝐴 ∈ V → (𝐴 ↾ 𝐵) ∈ V)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ↾ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2202  Vcvv 2803   ↾ cres 4733

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-sep 4212

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-res 4743

This theorem is referenced by:  sbthlemi10  7208  finomni  7382  ctinf  13112  znval  14712