Theorem aovvfunressn

Description: If the operation 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, 26-May-2017)

	Ref	Expression
Assertion	aovvfunressn	$$⊢((AFB))\in C\to \mathrm{Fun}\left({F\upharpoonright }_{\left\{⟨A,B⟩\right\}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-aov	$$⊢((AFB))=\left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)$$
2	1	eleq1i	$$⊢((AFB))\in C\leftrightarrow \left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)\in C$$
3		afvvfunressn	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)\in C\to \mathrm{Fun}\left({F\upharpoonright }_{\left\{⟨A,B⟩\right\}}\right)$$
4	2 3	sylbi	$$⊢((AFB))\in C\to \mathrm{Fun}\left({F\upharpoonright }_{\left\{⟨A,B⟩\right\}}\right)$$