Theorem ofexg

Description: A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014)

	Ref	Expression
Assertion	ofexg	⊢ ( 𝐴 ∈ 𝑉 → ( ∘f 𝑅 ↾ 𝐴 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		df-of	⊢ ∘f 𝑅 = ( 𝑓 ∈ V , 𝑔 ∈ V ↦ ( 𝑥 ∈ ( dom 𝑓 ∩ dom 𝑔 ) ↦ ( ( 𝑓 ‘ 𝑥 ) 𝑅 ( 𝑔 ‘ 𝑥 ) ) ) )
2	1	mpofun	⊢ Fun ∘f 𝑅
3		resfunexg	⊢ ( ( Fun ∘f 𝑅 ∧ 𝐴 ∈ 𝑉 ) → ( ∘f 𝑅 ↾ 𝐴 ) ∈ V )
4	2 3	mpan	⊢ ( 𝐴 ∈ 𝑉 → ( ∘f 𝑅 ↾ 𝐴 ) ∈ V )