Theorem fvprc

Description: A function's value at a proper class is the empty set. See fvprcALT for a proof that uses ax-pow instead of ax-pr . (Contributed by NM, 20-May-1998) Avoid ax-pow . (Revised by BTernaryTau, 3-Aug-2024) (Proof shortened by BTernaryTau, 3-Dec-2024)

	Ref	Expression
Assertion	fvprc	$$⊢\neg A\in \mathrm{V}\to F\left(A\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		brprcneu	$$⊢\neg A\in \mathrm{V}\to \neg \exists !xAFx$$
2		tz6.12-2	$$⊢\neg \exists !xAFx\to F\left(A\right)=\varnothing$$
3	1 2	syl	$$⊢\neg A\in \mathrm{V}\to F\left(A\right)=\varnothing$$