Theorem tpstop

Description: The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014)

	Ref	Expression
Hypothesis	tpstop.j	$$⊢J=\mathrm{TopOpen}\left(K\right)$$
Assertion	tpstop	$$⊢K\in \mathrm{TopSp}\to J\in \mathrm{Top}$$

Proof

Step	Hyp	Ref	Expression
1		tpstop.j	$$⊢J=\mathrm{TopOpen}\left(K\right)$$
2		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
3	2 1	istps2	$$⊢K\in \mathrm{TopSp}\leftrightarrow \left(J\in \mathrm{Top}\wedge {\mathrm{Base}}_K=\bigcup J\right)$$
4	3	simplbi	$$⊢K\in \mathrm{TopSp}\to J\in \mathrm{Top}$$