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 = ( MetOpen ‘ 𝐷 )
Assertion	mopnuni	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 )

Proof

Step	Hyp	Ref	Expression
1		mopnval.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2	1	mopntopon	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
3		toponuni	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 )
4	2 3	syl	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 )