Theorem hmeocnvcn

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

	Ref	Expression
Assertion	hmeocnvcn	⊢ ( 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) → ◡ 𝐹 ∈ ( 𝐾 Cn 𝐽 ) )

Proof

Step	Hyp	Ref	Expression
1		ishmeo	⊢ ( 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) ↔ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ ◡ 𝐹 ∈ ( 𝐾 Cn 𝐽 ) ) )
2	1	simprbi	⊢ ( 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) → ◡ 𝐹 ∈ ( 𝐾 Cn 𝐽 ) )