Theorem nfunsnaov

Description: If the restriction of a class to a singleton is not a function, its operation value is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017)

	Ref	Expression
Assertion	nfunsnaov	$$⊢\neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{⟨A,B⟩\right\}}\right)\to ((AFB))=\mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		df-aov	$$⊢((AFB))=\left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)$$
2		nfunsnafv	$$⊢\neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{⟨A,B⟩\right\}}\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)=\mathrm{V}$$
3	1 2	syl5eq	$$⊢\neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{⟨A,B⟩\right\}}\right)\to ((AFB))=\mathrm{V}$$