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 ) ) |
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 ) ) |