Theorem flimfcls

Description: A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009) (Revised by Stefan O'Rear, 8-Aug-2015)

	Ref	Expression
Assertion	flimfcls	$$⊢J\mathrm{fLim}F\subseteq J\mathrm{fClus}F$$

Proof

Step	Hyp	Ref	Expression
1		flimtop	$$⊢a\in \left(J\mathrm{fLim}F\right)\to J\in \mathrm{Top}$$
2		eqid	$$⊢\bigcup J=\bigcup J$$
3	2	flimfil	$$⊢a\in \left(J\mathrm{fLim}F\right)\to F\in \mathrm{Fil}\left(\bigcup J\right)$$
4		flimclsi	$$⊢x\in F\to J\mathrm{fLim}F\subseteq \mathrm{cls}\left(J\right)\left(x\right)$$
5	4	sseld	$$⊢x\in F\to \left(a\in \left(J\mathrm{fLim}F\right)\to a\in \mathrm{cls}\left(J\right)\left(x\right)\right)$$
6	5	com12	$$⊢a\in \left(J\mathrm{fLim}F\right)\to \left(x\in F\to a\in \mathrm{cls}\left(J\right)\left(x\right)\right)$$
7	6	ralrimiv	$$⊢a\in \left(J\mathrm{fLim}F\right)\to \forall x\in Fa\in \mathrm{cls}\left(J\right)\left(x\right)$$
8	2	isfcls	$$⊢a\in \left(J\mathrm{fClus}F\right)\leftrightarrow \left(J\in \mathrm{Top}\wedge F\in \mathrm{Fil}\left(\bigcup J\right)\wedge \forall x\in Fa\in \mathrm{cls}\left(J\right)\left(x\right)\right)$$
9	1 3 7 8	syl3anbrc	$$⊢a\in \left(J\mathrm{fLim}F\right)\to a\in \left(J\mathrm{fClus}F\right)$$
10	9	ssriv	$$⊢J\mathrm{fLim}F\subseteq J\mathrm{fClus}F$$