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	$$⊢\phi \to X={\mathrm{Base}}_M$$
	setsms.d	$$⊢\phi \to D={\mathrm{dist}\left(M\right)\upharpoonright }_{\left(X\times X\right)}$$
	setsms.k	$$⊢\phi \to K=M\mathrm{sSet}⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(D\right)⟩$$
Assertion	setsmsbas	$$⊢\phi \to X={\mathrm{Base}}_K$$

Proof

Step	Hyp	Ref	Expression
1		setsms.x	$$⊢\phi \to X={\mathrm{Base}}_M$$
2		setsms.d	$$⊢\phi \to D={\mathrm{dist}\left(M\right)\upharpoonright }_{\left(X\times X\right)}$$
3		setsms.k	$$⊢\phi \to K=M\mathrm{sSet}⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(D\right)⟩$$
4		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
5		tsetndxnbasendx	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
6	5	necomi	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{TopSet}\left(\mathrm{ndx}\right)$$
7	4 6	setsnid	$$⊢{\mathrm{Base}}_M={\mathrm{Base}}_{\left(M\mathrm{sSet}⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(D\right)⟩\right)}$$
8	3	fveq2d	$$⊢\phi \to {\mathrm{Base}}_K={\mathrm{Base}}_{\left(M\mathrm{sSet}⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(D\right)⟩\right)}$$
9	7 1 8	3eqtr4a	$$⊢\phi \to X={\mathrm{Base}}_K$$