Theorem nfunsnafv

Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn . (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	nfunsnafv	$$⊢\neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=\mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		df-dfat	$$⊢F\mathrm{defAt}A\leftrightarrow \left(A\in \mathrm{dom}F\wedge \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\right)$$
2	1	simprbi	$$⊢F\mathrm{defAt}A\to \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)$$
3		afvnfundmuv	$$⊢\neg F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=\mathrm{V}$$
4	2 3	nsyl5	$$⊢\neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=\mathrm{V}$$