Theorem eltpsg

Description: Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015) (Revised by AV, 31-Oct-2024)

	Ref	Expression
Hypothesis	eltpsi.k	$$⊢K=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},A⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}$$
Assertion	eltpsg	$$⊢J\in \mathrm{TopOn}\left(A\right)\to K\in \mathrm{TopSp}$$

Proof

Step	Hyp	Ref	Expression
1		eltpsi.k	$$⊢K=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},A⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}$$
2		basendxlttsetndx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}<\mathrm{TopSet}\left(\mathrm{ndx}\right)$$
3		tsetndxnn	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\in \mathbb{N}$$
4		tsetid	$$⊢\mathrm{TopSet}=\mathrm{Slot}\mathrm{TopSet}\left(\mathrm{ndx}\right)$$
5	1 2 3 4	2strop	$$⊢J\in \mathrm{TopOn}\left(A\right)\to J=\mathrm{TopSet}\left(K\right)$$
6		toponmax	$$⊢J\in \mathrm{TopOn}\left(A\right)\to A\in J$$
7	1 2 3	2strbas	$$⊢A\in J\to A={\mathrm{Base}}_K$$
8	6 7	syl	$$⊢J\in \mathrm{TopOn}\left(A\right)\to A={\mathrm{Base}}_K$$
9	8	fveq2d	$$⊢J\in \mathrm{TopOn}\left(A\right)\to \mathrm{TopOn}\left(A\right)=\mathrm{TopOn}\left({\mathrm{Base}}_K\right)$$
10	5 9	eleq12d	$$⊢J\in \mathrm{TopOn}\left(A\right)\to \left(J\in \mathrm{TopOn}\left(A\right)\leftrightarrow \mathrm{TopSet}\left(K\right)\in \mathrm{TopOn}\left({\mathrm{Base}}_K\right)\right)$$
11	10	ibi	$$⊢J\in \mathrm{TopOn}\left(A\right)\to \mathrm{TopSet}\left(K\right)\in \mathrm{TopOn}\left({\mathrm{Base}}_K\right)$$
12		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
13		eqid	$$⊢\mathrm{TopSet}\left(K\right)=\mathrm{TopSet}\left(K\right)$$
14	12 13	tsettps	$$⊢\mathrm{TopSet}\left(K\right)\in \mathrm{TopOn}\left({\mathrm{Base}}_K\right)\to K\in \mathrm{TopSp}$$
15	11 14	syl	$$⊢J\in \mathrm{TopOn}\left(A\right)\to K\in \mathrm{TopSp}$$