Theorem eltpsi

Description: Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012) (Revised by Mario Carneiro, 13-Aug-2015)

	Ref	Expression
Hypotheses	eltpsi.k	$$⊢K=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},A⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}$$
	eltpsi.u	$$⊢A=\bigcup J$$
	eltpsi.j	$$⊢J\in \mathrm{Top}$$
Assertion	eltpsi	$$⊢K\in \mathrm{TopSp}$$

Proof

Step	Hyp	Ref	Expression
1		eltpsi.k	$$⊢K=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},A⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}$$
2		eltpsi.u	$$⊢A=\bigcup J$$
3		eltpsi.j	$$⊢J\in \mathrm{Top}$$
4	2	toptopon	$$⊢J\in \mathrm{Top}\leftrightarrow J\in \mathrm{TopOn}\left(A\right)$$
5	3 4	mpbi	$$⊢J\in \mathrm{TopOn}\left(A\right)$$
6	1	eltpsg	$$⊢J\in \mathrm{TopOn}\left(A\right)\to K\in \mathrm{TopSp}$$
7	5 6	ax-mp	$$⊢K\in \mathrm{TopSp}$$