Theorem afvvdm

Description: If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	afvvdm	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B\to A\in \mathrm{dom}F$$

Proof

Step	Hyp	Ref	Expression
1		ndmafv	$$⊢\neg A\in \mathrm{dom}F\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=\mathrm{V}$$
2		nvelim	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=\mathrm{V}\to \neg \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B$$
3	1 2	syl	$$⊢\neg A\in \mathrm{dom}F\to \neg \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B$$
4	3	con4i	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B\to A\in \mathrm{dom}F$$