Theorem cmetmet

Description: A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006) (Revised by Mario Carneiro, 29-Jan-2014)

	Ref	Expression
Assertion	cmetmet	⊢ ( 𝐷 ∈ ( CMet ‘ 𝑋 ) → 𝐷 ∈ ( Met ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( MetOpen ‘ 𝐷 ) = ( MetOpen ‘ 𝐷 )
2	1	iscmet	⊢ ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ↔ ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ∀ 𝑓 ∈ ( CauFil ‘ 𝐷 ) ( ( MetOpen ‘ 𝐷 ) fLim 𝑓 ) ≠ ∅ ) )
3	2	simplbi	⊢ ( 𝐷 ∈ ( CMet ‘ 𝑋 ) → 𝐷 ∈ ( Met ‘ 𝑋 ) )