Theorem setsmsbas

Description: The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015) (Proof shortened by AV, 12-Nov-2024)

	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 ) ⟩ ) )
Assertion	setsmsbas	⊢ ( 𝜑 → 𝑋 = ( Base ‘ 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		baseid	⊢ Base = Slot ( Base ‘ ndx )
5		tsetndxnbasendx	⊢ ( TopSet ‘ ndx ) ≠ ( Base ‘ ndx )
6	5	necomi	⊢ ( Base ‘ ndx ) ≠ ( TopSet ‘ ndx )
7	4 6	setsnid	⊢ ( Base ‘ 𝑀 ) = ( Base ‘ ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) )
8	3	fveq2d	⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) ) )
9	7 1 8	3eqtr4a	⊢ ( 𝜑 → 𝑋 = ( Base ‘ 𝐾 ) )