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 = U. J
Assertion	qtoppconn	|- ( ( J e. PConn /\ F Fn X ) -> ( J qTop F ) e. PConn )

Proof

Step	Hyp	Ref	Expression
1		pconnpi1.x	|- X = U. J
2		pconntop	|- ( J e. PConn -> J e. Top )
3		eqid	|- U. ( J qTop F ) = U. ( J qTop F )
4	3	cnpconn	|- ( ( J e. PConn /\ F : X -onto-> U. ( J qTop F ) /\ F e. ( J Cn ( J qTop F ) ) ) -> ( J qTop F ) e. PConn )
5	1 2 4	qtopcmplem	|- ( ( J e. PConn /\ F Fn X ) -> ( J qTop F ) e. PConn )