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	$$⊢J\in \mathrm{TopOn}\left(X\right)\to J\in \mathrm{TopOn}\left(\bigcup J\right)$$

Proof

Step	Hyp	Ref	Expression
1		topontop	$$⊢J\in \mathrm{TopOn}\left(X\right)\to J\in \mathrm{Top}$$
2		toptopon2	$$⊢J\in \mathrm{Top}\leftrightarrow J\in \mathrm{TopOn}\left(\bigcup J\right)$$
3	1 2	sylib	$$⊢J\in \mathrm{TopOn}\left(X\right)\to J\in \mathrm{TopOn}\left(\bigcup J\right)$$