Theorem qtopconn

Description: A quotient of a connected space is connected. (Contributed by Mario Carneiro, 24-Mar-2015)

	Ref	Expression
Hypothesis	qtopcmp.1	$$⊢X=\bigcup J$$
Assertion	qtopconn	$$⊢\left(J\in \mathrm{Conn}\wedge FFnX\right)\to J\mathrm{qTop}F\in \mathrm{Conn}$$

Proof

Step	Hyp	Ref	Expression
1		qtopcmp.1	$$⊢X=\bigcup J$$
2		conntop	$$⊢J\in \mathrm{Conn}\to J\in \mathrm{Top}$$
3		eqid	$$⊢\bigcup \left(J\mathrm{qTop}F\right)=\bigcup \left(J\mathrm{qTop}F\right)$$
4	3	cnconn	$$⊢\left(J\in \mathrm{Conn}\wedge F:X\underset{onto}{⟶}\bigcup \left(J\mathrm{qTop}F\right)\wedge F\in \left(J\mathrm{Cn}\left(J\mathrm{qTop}F\right)\right)\right)\to J\mathrm{qTop}F\in \mathrm{Conn}$$
5	1 2 4	qtopcmplem	$$⊢\left(J\in \mathrm{Conn}\wedge FFnX\right)\to J\mathrm{qTop}F\in \mathrm{Conn}$$