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Metamath Proof Explorer

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 ) )