Theorem isconn

Description: The predicate J is a connected topology . (Contributed by FL, 17-Nov-2008)

	Ref	Expression
Hypothesis	isconn.1	$$⊢X=\bigcup J$$
Assertion	isconn	$$⊢J\in \mathrm{Conn}\leftrightarrow \left(J\in \mathrm{Top}\wedge J\cap \mathrm{Clsd}\left(J\right)=\left\{\varnothing ,X\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		isconn.1	$$⊢X=\bigcup J$$
2		id	$$⊢j=J\to j=J$$
3		fveq2	$$⊢j=J\to \mathrm{Clsd}\left(j\right)=\mathrm{Clsd}\left(J\right)$$
4	2 3	ineq12d	$$⊢j=J\to j\cap \mathrm{Clsd}\left(j\right)=J\cap \mathrm{Clsd}\left(J\right)$$
5		unieq	$$⊢j=J\to \bigcup j=\bigcup J$$
6	5 1	eqtr4di	$$⊢j=J\to \bigcup j=X$$
7	6	preq2d	$$⊢j=J\to \left\{\varnothing ,\bigcup j\right\}=\left\{\varnothing ,X\right\}$$
8	4 7	eqeq12d	$$⊢j=J\to \left(j\cap \mathrm{Clsd}\left(j\right)=\left\{\varnothing ,\bigcup j\right\}\leftrightarrow J\cap \mathrm{Clsd}\left(J\right)=\left\{\varnothing ,X\right\}\right)$$
9		df-conn	$$⊢\mathrm{Conn}=\left\{j\in \mathrm{Top}|j\cap \mathrm{Clsd}\left(j\right)=\left\{\varnothing ,\bigcup j\right\}\right\}$$
10	8 9	elrab2	$$⊢J\in \mathrm{Conn}\leftrightarrow \left(J\in \mathrm{Top}\wedge J\cap \mathrm{Clsd}\left(J\right)=\left\{\varnothing ,X\right\}\right)$$