Theorem fnresin2

Description: Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994)

	Ref	Expression
Assertion	fnresin2	⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 ↾ ( 𝐵 ∩ 𝐴 ) ) Fn ( 𝐵 ∩ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		inss2	⊢ ( 𝐵 ∩ 𝐴 ) ⊆ 𝐴
2		fnssres	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( 𝐵 ∩ 𝐴 ) ⊆ 𝐴 ) → ( 𝐹 ↾ ( 𝐵 ∩ 𝐴 ) ) Fn ( 𝐵 ∩ 𝐴 ) )
3	1 2	mpan2	⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 ↾ ( 𝐵 ∩ 𝐴 ) ) Fn ( 𝐵 ∩ 𝐴 ) )