Theorem metdmdm

Description: Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015)

	Ref	Expression
Assertion	metdmdm	⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝑋 = dom dom 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		metxmet	⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
2		xmetdmdm	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = dom dom 𝐷 )
3	1 2	syl	⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝑋 = dom dom 𝐷 )