Theorem ovn0dmfun

Description: If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 . (Contributed by AV, 27-Jan-2020)

	Ref	Expression
Assertion	ovn0dmfun	$$⊢AFB\ne \varnothing \to \left(⟨A,B⟩\in \mathrm{dom}F\wedge \mathrm{Fun}\left({F\upharpoonright }_{\left\{⟨A,B⟩\right\}}\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-ov	$$⊢AFB=F\left(⟨A,B⟩\right)$$
2	1	neeq1i	$$⊢AFB\ne \varnothing \leftrightarrow F\left(⟨A,B⟩\right)\ne \varnothing$$
3		fvfundmfvn0	$$⊢F\left(⟨A,B⟩\right)\ne \varnothing \to \left(⟨A,B⟩\in \mathrm{dom}F\wedge \mathrm{Fun}\left({F\upharpoonright }_{\left\{⟨A,B⟩\right\}}\right)\right)$$
4	2 3	sylbi	$$⊢AFB\ne \varnothing \to \left(⟨A,B⟩\in \mathrm{dom}F\wedge \mathrm{Fun}\left({F\upharpoonright }_{\left\{⟨A,B⟩\right\}}\right)\right)$$