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	$$⊢\mathrm{Fun}F\to \mathrm{Fun}\left({F\upharpoonright }_A\right)$$

Proof

Step	Hyp	Ref	Expression
1		resss	$$⊢{F\upharpoonright }_A\subseteq F$$
2		funss	$$⊢{F\upharpoonright }_A\subseteq F\to \left(\mathrm{Fun}F\to \mathrm{Fun}\left({F\upharpoonright }_A\right)\right)$$
3	1 2	ax-mp	$$⊢\mathrm{Fun}F\to \mathrm{Fun}\left({F\upharpoonright }_A\right)$$