Theorem mapprc

Description: When A is a proper class, the class of all functions mapping A to B is empty. Exercise 4.41 of Mendelson p. 255. (Contributed by NM, 8-Dec-2003)

	Ref	Expression
Assertion	mapprc	$$⊢\neg A\in \mathrm{V}\to \left\{f|f:A⟶B\right\}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		abn0	$$⊢\left\{f|f:A⟶B\right\}\ne \varnothing \leftrightarrow \exists ff:A⟶B$$
2		fdm	$$⊢f:A⟶B\to \mathrm{dom}f=A$$
3		vex	$$⊢f\in \mathrm{V}$$
4	3	dmex	$$⊢\mathrm{dom}f\in \mathrm{V}$$
5	2 4	eqeltrrdi	$$⊢f:A⟶B\to A\in \mathrm{V}$$
6	5	exlimiv	$$⊢\exists ff:A⟶B\to A\in \mathrm{V}$$
7	1 6	sylbi	$$⊢\left\{f|f:A⟶B\right\}\ne \varnothing \to A\in \mathrm{V}$$
8	7	necon1bi	$$⊢\neg A\in \mathrm{V}\to \left\{f|f:A⟶B\right\}=\varnothing$$