Theorem kgenuni

Description: The base set of the compact generator is the same as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015)

	Ref	Expression
Hypothesis	kgenuni.1	$$⊢X=\bigcup J$$
Assertion	kgenuni	$$⊢J\in \mathrm{Top}\to X=\bigcup \mathrm{𝑘Gen}\left(J\right)$$

Proof

Step	Hyp	Ref	Expression
1		kgenuni.1	$$⊢X=\bigcup J$$
2	1	toptopon	$$⊢J\in \mathrm{Top}\leftrightarrow J\in \mathrm{TopOn}\left(X\right)$$
3		kgentopon	$$⊢J\in \mathrm{TopOn}\left(X\right)\to \mathrm{𝑘Gen}\left(J\right)\in \mathrm{TopOn}\left(X\right)$$
4	2 3	sylbi	$$⊢J\in \mathrm{Top}\to \mathrm{𝑘Gen}\left(J\right)\in \mathrm{TopOn}\left(X\right)$$
5		toponuni	$$⊢\mathrm{𝑘Gen}\left(J\right)\in \mathrm{TopOn}\left(X\right)\to X=\bigcup \mathrm{𝑘Gen}\left(J\right)$$
6	4 5	syl	$$⊢J\in \mathrm{Top}\to X=\bigcup \mathrm{𝑘Gen}\left(J\right)$$