Theorem qtopconn

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

	Ref	Expression
Hypothesis	qtopcmp.1	|- X = U. J
Assertion	qtopconn	|- ( ( J e. Conn /\ F Fn X ) -> ( J qTop F ) e. Conn )

Proof

Step	Hyp	Ref	Expression
1		qtopcmp.1	|- X = U. J
2		conntop	|- ( J e. Conn -> J e. Top )
3		eqid	|- U. ( J qTop F ) = U. ( J qTop F )
4	3	cnconn	|- ( ( J e. Conn /\ F : X -onto-> U. ( J qTop F ) /\ F e. ( J Cn ( J qTop F ) ) ) -> ( J qTop F ) e. Conn )
5	1 2 4	qtopcmplem	|- ( ( J e. Conn /\ F Fn X ) -> ( J qTop F ) e. Conn )