Theorem cntop1

Description: Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015)

	Ref	Expression
Assertion	cntop1	$$⊢F\in \left(J\mathrm{Cn}K\right)\to J\in \mathrm{Top}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\bigcup J=\bigcup J$$
2		eqid	$$⊢\bigcup K=\bigcup K$$
3	1 2	iscn2	$$⊢F\in \left(J\mathrm{Cn}K\right)\leftrightarrow \left(\left(J\in \mathrm{Top}\wedge K\in \mathrm{Top}\right)\wedge \left(F:\bigcup J⟶\bigcup K\wedge \forall x\in K{F}^{-1}\left[x\right]\in J\right)\right)$$
4	3	simplbi	$$⊢F\in \left(J\mathrm{Cn}K\right)\to \left(J\in \mathrm{Top}\wedge K\in \mathrm{Top}\right)$$
5	4	simpld	$$⊢F\in \left(J\mathrm{Cn}K\right)\to J\in \mathrm{Top}$$