Theorem topfne

Description: Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009)

	Ref	Expression
Hypotheses	topfne.1	$$⊢X=\bigcup J$$
	topfne.2	$$⊢Y=\bigcup K$$
Assertion	topfne	$$⊢\left(K\in \mathrm{Top}\wedge X=Y\right)\to \left(J\subseteq K\leftrightarrow J\mathrm{Fne}K\right)$$

Proof

Step	Hyp	Ref	Expression
1		topfne.1	$$⊢X=\bigcup J$$
2		topfne.2	$$⊢Y=\bigcup K$$
3		tgtop	$$⊢K\in \mathrm{Top}\to \mathrm{topGen}\left(K\right)=K$$
4	3	sseq2d	$$⊢K\in \mathrm{Top}\to \left(J\subseteq \mathrm{topGen}\left(K\right)\leftrightarrow J\subseteq K\right)$$
5	4	bicomd	$$⊢K\in \mathrm{Top}\to \left(J\subseteq K\leftrightarrow J\subseteq \mathrm{topGen}\left(K\right)\right)$$
6	1 2	isfne4	$$⊢J\mathrm{Fne}K\leftrightarrow \left(X=Y\wedge J\subseteq \mathrm{topGen}\left(K\right)\right)$$
7	6	baibr	$$⊢X=Y\to \left(J\subseteq \mathrm{topGen}\left(K\right)\leftrightarrow J\mathrm{Fne}K\right)$$
8	5 7	sylan9bb	$$⊢\left(K\in \mathrm{Top}\wedge X=Y\right)\to \left(J\subseteq K\leftrightarrow J\mathrm{Fne}K\right)$$