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 = ( TopOpen ‘ 𝐾 )
Assertion	tpstop	⊢ ( 𝐾 ∈ TopSp → 𝐽 ∈ Top )

Proof

Step	Hyp	Ref	Expression
1		tpstop.j	⊢ 𝐽 = ( TopOpen ‘ 𝐾 )
2		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
3	2 1	istps2	⊢ ( 𝐾 ∈ TopSp ↔ ( 𝐽 ∈ Top ∧ ( Base ‘ 𝐾 ) = ∪ 𝐽 ) )
4	3	simplbi	⊢ ( 𝐾 ∈ TopSp → 𝐽 ∈ Top )