This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )