Metamath Proof Explorer


Theorem tmsbas

Description: The base set of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015)

Ref Expression
Hypothesis tmsbas.k 𝐾 = ( toMetSp ‘ 𝐷 )
Assertion tmsbas ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = ( Base ‘ 𝐾 ) )

Proof

Step Hyp Ref Expression
1 tmsbas.k 𝐾 = ( toMetSp ‘ 𝐷 )
2 eqid { ⟨ ( Base ‘ ndx ) , 𝑋 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } = { ⟨ ( Base ‘ ndx ) , 𝑋 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ }
3 2 1 tmslem ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝑋 = ( Base ‘ 𝐾 ) ∧ 𝐷 = ( dist ‘ 𝐾 ) ∧ ( MetOpen ‘ 𝐷 ) = ( TopOpen ‘ 𝐾 ) ) )
4 3 simp1d ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = ( Base ‘ 𝐾 ) )