Theorem fununfun

Description: If the union of classes is a function, the classes themselves are functions. (Contributed by AV, 18-Jul-2019) Shorten proof and remove dependency on ax-sep and ax-pr . (Revised by Eric Schmidt, 24-Oct-2026)

	Ref	Expression
Assertion	fununfun	$$⊢\mathrm{Fun}\left(F\cup G\right)\to \left(\mathrm{Fun}F\wedge \mathrm{Fun}G\right)$$

Proof

Step	Hyp	Ref	Expression
1		ssun1	$$⊢F\subseteq F\cup G$$
2		funss	$$⊢F\subseteq F\cup G\to \left(\mathrm{Fun}\left(F\cup G\right)\to \mathrm{Fun}F\right)$$
3	1 2	ax-mp	$$⊢\mathrm{Fun}\left(F\cup G\right)\to \mathrm{Fun}F$$
4		ssun2	$$⊢G\subseteq F\cup G$$
5		funss	$$⊢G\subseteq F\cup G\to \left(\mathrm{Fun}\left(F\cup G\right)\to \mathrm{Fun}G\right)$$
6	4 5	ax-mp	$$⊢\mathrm{Fun}\left(F\cup G\right)\to \mathrm{Fun}G$$
7	3 6	jca	$$⊢\mathrm{Fun}\left(F\cup G\right)\to \left(\mathrm{Fun}F\wedge \mathrm{Fun}G\right)$$