Theorem topontopn

Description: Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015)

	Ref	Expression
Hypotheses	tsettps.a	⊢ 𝐴 = ( Base ‘ 𝐾 )
	tsettps.j	⊢ 𝐽 = ( TopSet ‘ 𝐾 )
Assertion	topontopn	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐴 ) → 𝐽 = ( TopOpen ‘ 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		tsettps.a	⊢ 𝐴 = ( Base ‘ 𝐾 )
2		tsettps.j	⊢ 𝐽 = ( TopSet ‘ 𝐾 )
3		toponuni	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐴 ) → 𝐴 = ∪ 𝐽 )
4		eqimss2	⊢ ( 𝐴 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝐴 )
5	3 4	syl	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐴 ) → ∪ 𝐽 ⊆ 𝐴 )
6		sspwuni	⊢ ( 𝐽 ⊆ 𝒫 𝐴 ↔ ∪ 𝐽 ⊆ 𝐴 )
7	5 6	sylibr	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐴 ) → 𝐽 ⊆ 𝒫 𝐴 )
8	1 2	topnid	⊢ ( 𝐽 ⊆ 𝒫 𝐴 → 𝐽 = ( TopOpen ‘ 𝐾 ) )
9	7 8	syl	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐴 ) → 𝐽 = ( TopOpen ‘ 𝐾 ) )