Theorem flimtop

Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015) (Revised by Stefan O'Rear, 9-Aug-2015)

	Ref	Expression
Assertion	flimtop	$$⊢A\in \left(J\mathrm{fLim}F\right)\to J\in \mathrm{Top}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\bigcup J=\bigcup J$$
2	1	elflim2	$$⊢A\in \left(J\mathrm{fLim}F\right)\leftrightarrow \left(\left(J\in \mathrm{Top}\wedge F\in \bigcup \mathrm{ran}\mathrm{Fil}\wedge F\subseteq 𝒫\bigcup J\right)\wedge \left(A\in \bigcup J\wedge \mathrm{nei}\left(J\right)\left(\left\{A\right\}\right)\subseteq F\right)\right)$$
3	2	simplbi	$$⊢A\in \left(J\mathrm{fLim}F\right)\to \left(J\in \mathrm{Top}\wedge F\in \bigcup \mathrm{ran}\mathrm{Fil}\wedge F\subseteq 𝒫\bigcup J\right)$$
4	3	simp1d	$$⊢A\in \left(J\mathrm{fLim}F\right)\to J\in \mathrm{Top}$$