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 ∈ ( TopOn ‘ 𝐴 ) → 𝐾 ∈ TopSp )

Proof

Step	Hyp	Ref	Expression
1		tsettps.a	⊢ 𝐴 = ( Base ‘ 𝐾 )
2		tsettps.j	⊢ 𝐽 = ( TopSet ‘ 𝐾 )
3	1 2	topontopn	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐴 ) → 𝐽 = ( TopOpen ‘ 𝐾 ) )
4		id	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐴 ) → 𝐽 ∈ ( TopOn ‘ 𝐴 ) )
5	3 4	eqeltrrd	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐴 ) → ( TopOpen ‘ 𝐾 ) ∈ ( TopOn ‘ 𝐴 ) )
6		eqid	⊢ ( TopOpen ‘ 𝐾 ) = ( TopOpen ‘ 𝐾 )
7	1 6	istps	⊢ ( 𝐾 ∈ TopSp ↔ ( TopOpen ‘ 𝐾 ) ∈ ( TopOn ‘ 𝐴 ) )
8	5 7	sylibr	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐴 ) → 𝐾 ∈ TopSp )