Theorem qtoppconn

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

	Ref	Expression
Hypothesis	pconnpi1.x	$$⊢X=\bigcup J$$
Assertion	qtoppconn	$$⊢\left(J\in \mathrm{PConn}\wedge FFnX\right)\to J\mathrm{qTop}F\in \mathrm{PConn}$$

Proof

Step	Hyp	Ref	Expression
1		pconnpi1.x	$$⊢X=\bigcup J$$
2		pconntop	$$⊢J\in \mathrm{PConn}\to J\in \mathrm{Top}$$
3		eqid	$$⊢\bigcup \left(J\mathrm{qTop}F\right)=\bigcup \left(J\mathrm{qTop}F\right)$$
4	3	cnpconn	$$⊢\left(J\in \mathrm{PConn}\wedge F:X\underset{onto}{⟶}\bigcup \left(J\mathrm{qTop}F\right)\wedge F\in \left(J\mathrm{Cn}\left(J\mathrm{qTop}F\right)\right)\right)\to J\mathrm{qTop}F\in \mathrm{PConn}$$
5	1 2 4	qtopcmplem	$$⊢\left(J\in \mathrm{PConn}\wedge FFnX\right)\to J\mathrm{qTop}F\in \mathrm{PConn}$$