Theorem topmtcl

Description: The meet of a collection of topologies on X is again a topology on X . (Contributed by Jeff Hankins, 5-Oct-2009) (Proof shortened by Mario Carneiro, 12-Sep-2015)

	Ref	Expression
Assertion	topmtcl	$$⊢\left(X\in V\wedge S\subseteq \mathrm{TopOn}\left(X\right)\right)\to 𝒫X\cap \bigcap S\in \mathrm{TopOn}\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		toponmre	$$⊢X\in V\to \mathrm{TopOn}\left(X\right)\in \mathrm{Moore}\left(𝒫X\right)$$
2		mrerintcl	$$⊢\left(\mathrm{TopOn}\left(X\right)\in \mathrm{Moore}\left(𝒫X\right)\wedge S\subseteq \mathrm{TopOn}\left(X\right)\right)\to 𝒫X\cap \bigcap S\in \mathrm{TopOn}\left(X\right)$$
3	1 2	sylan	$$⊢\left(X\in V\wedge S\subseteq \mathrm{TopOn}\left(X\right)\right)\to 𝒫X\cap \bigcap S\in \mathrm{TopOn}\left(X\right)$$