Theorem topontopon

Description: A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021)

	Ref	Expression
Assertion	topontopon	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) )

Proof

Step	Hyp	Ref	Expression
1		topontop	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top )
2		toptopon2	⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) )
3	1 2	sylib	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) )