Theorem hmeocn

Description: A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015)

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

Proof

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