Theorem hmeocnvcn

Description: The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015)

	Ref	Expression
Assertion	hmeocnvcn	$$⊢F\in \left(J\mathrm{Homeo}K\right)\to F^{-1}\in \left(K\mathrm{Cn}J\right)$$

Proof

Step	Hyp	Ref	Expression
1		ishmeo	$$⊢F\in \left(J\mathrm{Homeo}K\right)\leftrightarrow \left(F\in \left(J\mathrm{Cn}K\right)\wedge F^{-1}\in \left(K\mathrm{Cn}J\right)\right)$$
2	1	simprbi	$$⊢F\in \left(J\mathrm{Homeo}K\right)\to F^{-1}\in \left(K\mathrm{Cn}J\right)$$