Theorem resexg

Description: The restriction of a set is a set. (Contributed by NM, 28-Mar-1998) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	resexg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ↾ 𝐵 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		resss	⊢ ( 𝐴 ↾ 𝐵 ) ⊆ 𝐴
2		ssexg	⊢ ( ( ( 𝐴 ↾ 𝐵 ) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ↾ 𝐵 ) ∈ V )
3	1 2	mpan	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ↾ 𝐵 ) ∈ V )