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Metamath Proof Explorer

Theorem setsmstset

Description: The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015)

Ref Expression
Hypotheses setsms.x ⊢ ( 𝜑 → 𝑋 = ( Base ‘ 𝑀 ) )
setsms.d ⊢ ( 𝜑 → 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) )
setsms.k ⊢ ( 𝜑 → 𝐾 = ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) )
setsms.m ⊢ ( 𝜑 → 𝑀 ∈ 𝑉 )
Assertion setsmstset ( 𝜑 → ( MetOpen ‘ 𝐷 ) = ( TopSet ‘ 𝐾 ) )

Proof

Step Hyp Ref Expression
1 setsms.x ⊢ ( 𝜑 → 𝑋 = ( Base ‘ 𝑀 ) )
2 setsms.d ⊢ ( 𝜑 → 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) )
3 setsms.k ⊢ ( 𝜑 → 𝐾 = ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) )
4 setsms.m ⊢ ( 𝜑 → 𝑀 ∈ 𝑉 )
5 fvex ⊢ ( MetOpen ‘ 𝐷 ) ∈ V
6 tsetid ⊢ TopSet = Slot ( TopSet ‘ ndx )
7 6 setsid ⊢ ( ( 𝑀 ∈ 𝑉 ∧ ( MetOpen ‘ 𝐷 ) ∈ V ) → ( MetOpen ‘ 𝐷 ) = ( TopSet ‘ ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) ) )
8 4 5 7 sylancl ⊢ ( 𝜑 → ( MetOpen ‘ 𝐷 ) = ( TopSet ‘ ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) ) )
9 3 fveq2d ⊢ ( 𝜑 → ( TopSet ‘ 𝐾 ) = ( TopSet ‘ ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) ) )
10 8 9 eqtr4d ⊢ ( 𝜑 → ( MetOpen ‘ 𝐷 ) = ( TopSet ‘ 𝐾 ) )