Theorem fsetdmprc0

Description: The set of functions with a proper class as domain is empty. (Contributed by AV, 22-Aug-2024)

	Ref	Expression
Assertion	fsetdmprc0	$$⊢A\notin \mathrm{V}\to \left\{f|fFnA\right\}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		df-nel	$$⊢A\notin \mathrm{V}\leftrightarrow \neg A\in \mathrm{V}$$
2		vex	$$⊢f\in \mathrm{V}$$
3	2	a1i	$$⊢fFnA\to f\in \mathrm{V}$$
4		id	$$⊢fFnA\to fFnA$$
5	3 4	fndmexd	$$⊢fFnA\to A\in \mathrm{V}$$
6	5	con3i	$$⊢\neg A\in \mathrm{V}\to \neg fFnA$$
7	1 6	sylbi	$$⊢A\notin \mathrm{V}\to \neg fFnA$$
8	7	alrimiv	$$⊢A\notin \mathrm{V}\to \forall f\neg fFnA$$
9		ab0	$$⊢\left\{f|fFnA\right\}=\varnothing \leftrightarrow \forall f\neg fFnA$$
10	8 9	sylibr	$$⊢A\notin \mathrm{V}\to \left\{f|fFnA\right\}=\varnothing$$