Theorem dmresexg

Description: The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995)

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

Proof

Step	Hyp	Ref	Expression
1		dmres	⊢ dom ( 𝐴 ↾ 𝐵 ) = ( 𝐵 ∩ dom 𝐴 )
2		inex1g	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∩ dom 𝐴 ) ∈ V )
3	1 2	eqeltrid	⊢ ( 𝐵 ∈ 𝑉 → dom ( 𝐴 ↾ 𝐵 ) ∈ V )