Theorem qtopcmp

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

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

Proof

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