Theorem funres

Description: A restriction of a function is a function. Compare Exercise 18 of TakeutiZaring p. 25. (Contributed by NM, 16-Aug-1994)

	Ref	Expression
Assertion	funres	⊢ ( Fun 𝐹 → Fun ( 𝐹 ↾ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		resss	⊢ ( 𝐹 ↾ 𝐴 ) ⊆ 𝐹
2		funss	⊢ ( ( 𝐹 ↾ 𝐴 ) ⊆ 𝐹 → ( Fun 𝐹 → Fun ( 𝐹 ↾ 𝐴 ) ) )
3	1 2	ax-mp	⊢ ( Fun 𝐹 → Fun ( 𝐹 ↾ 𝐴 ) )