Theorem mopnuni

Description: The union of all open sets in a metric space is its underlying set. (Contributed by NM, 4-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)

	Ref	Expression
Hypothesis	mopnval.1	$$⊢J=\mathrm{MetOpen}\left(D\right)$$
Assertion	mopnuni	$$⊢D\in \mathrm{\infty Met}\left(X\right)\to X=\bigcup J$$

Proof

Step	Hyp	Ref	Expression
1		mopnval.1	$$⊢J=\mathrm{MetOpen}\left(D\right)$$
2	1	mopntopon	$$⊢D\in \mathrm{\infty Met}\left(X\right)\to J\in \mathrm{TopOn}\left(X\right)$$
3		toponuni	$$⊢J\in \mathrm{TopOn}\left(X\right)\to X=\bigcup J$$
4	2 3	syl	$$⊢D\in \mathrm{\infty Met}\left(X\right)\to X=\bigcup J$$