Theorem afvvfunressn

Description: If the function value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	afvvfunressn	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B\to \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfunsnafv	$$⊢\neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=\mathrm{V}$$
2		nvelim	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=\mathrm{V}\to \neg \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B$$
3	1 2	syl	$$⊢\neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\to \neg \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B$$
4	3	con4i	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B\to \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)$$