Theorem tsettps

Description: If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015)

	Ref	Expression
Hypotheses	tsettps.a	|- A = ( Base ` K )
	tsettps.j	|- J = ( TopSet ` K )
Assertion	tsettps	|- ( J e. ( TopOn ` A ) -> K e. TopSp )

Proof

Step	Hyp	Ref	Expression
1		tsettps.a	|- A = ( Base ` K )
2		tsettps.j	|- J = ( TopSet ` K )
3	1 2	topontopn	|- ( J e. ( TopOn ` A ) -> J = ( TopOpen ` K ) )
4		id	|- ( J e. ( TopOn ` A ) -> J e. ( TopOn ` A ) )
5	3 4	eqeltrrd	|- ( J e. ( TopOn ` A ) -> ( TopOpen ` K ) e. ( TopOn ` A ) )
6		eqid	|- ( TopOpen ` K ) = ( TopOpen ` K )
7	1 6	istps	|- ( K e. TopSp <-> ( TopOpen ` K ) e. ( TopOn ` A ) )
8	5 7	sylibr	|- ( J e. ( TopOn ` A ) -> K e. TopSp )