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 ) )