Theorem aovvdm

Description: If the operation value of a class for an ordered pair is a set, the ordered pair is contained in the domain of the class. (Contributed by Alexander van der Vekens, 26-May-2017)

	Ref	Expression
Assertion	aovvdm	$$⊢((AFB))\in C\to ⟨A,B⟩\in \mathrm{dom}F$$

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		afvvdm	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)\in C\to ⟨A,B⟩\in \mathrm{dom}F$$
4	2 3	sylbi	$$⊢((AFB))\in C\to ⟨A,B⟩\in \mathrm{dom}F$$