Theorem topontop

Description: A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015)

	Ref	Expression
Assertion	topontop	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐵 ) → 𝐽 ∈ Top )

Proof

Step	Hyp	Ref	Expression
1		istopon	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐵 ) ↔ ( 𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽 ) )
2	1	simplbi	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐵 ) → 𝐽 ∈ Top )