Theorem dmv

Description: The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003)

	Ref	Expression
Assertion	dmv	$$⊢\mathrm{dom}\mathrm{V}=\mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		ssv	$$⊢\mathrm{dom}\mathrm{V}\subseteq \mathrm{V}$$
2		dmi	$$⊢\mathrm{dom}\mathrm{I}=\mathrm{V}$$
3		ssv	$$⊢\mathrm{I}\subseteq \mathrm{V}$$
4		dmss	$$⊢\mathrm{I}\subseteq \mathrm{V}\to \mathrm{dom}\mathrm{I}\subseteq \mathrm{dom}\mathrm{V}$$
5	3 4	ax-mp	$$⊢\mathrm{dom}\mathrm{I}\subseteq \mathrm{dom}\mathrm{V}$$
6	2 5	eqsstrri	$$⊢\mathrm{V}\subseteq \mathrm{dom}\mathrm{V}$$
7	1 6	eqssi	$$⊢\mathrm{dom}\mathrm{V}=\mathrm{V}$$