Theorem hmeocnvb

Description: The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007) (Revised by Mario Carneiro, 23-Aug-2015)

	Ref	Expression
Assertion	hmeocnvb	⊢ ( Rel 𝐹 → ( ◡ 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) ↔ 𝐹 ∈ ( 𝐾 Homeo 𝐽 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hmeocnv	⊢ ( ◡ 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) → ◡ ◡ 𝐹 ∈ ( 𝐾 Homeo 𝐽 ) )
2		dfrel2	⊢ ( Rel 𝐹 ↔ ◡ ◡ 𝐹 = 𝐹 )
3		eleq1	⊢ ( ◡ ◡ 𝐹 = 𝐹 → ( ◡ ◡ 𝐹 ∈ ( 𝐾 Homeo 𝐽 ) ↔ 𝐹 ∈ ( 𝐾 Homeo 𝐽 ) ) )
4	2 3	sylbi	⊢ ( Rel 𝐹 → ( ◡ ◡ 𝐹 ∈ ( 𝐾 Homeo 𝐽 ) ↔ 𝐹 ∈ ( 𝐾 Homeo 𝐽 ) ) )
5	1 4	imbitrid	⊢ ( Rel 𝐹 → ( ◡ 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) → 𝐹 ∈ ( 𝐾 Homeo 𝐽 ) ) )
6		hmeocnv	⊢ ( 𝐹 ∈ ( 𝐾 Homeo 𝐽 ) → ◡ 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) )
7	5 6	impbid1	⊢ ( Rel 𝐹 → ( ◡ 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) ↔ 𝐹 ∈ ( 𝐾 Homeo 𝐽 ) ) )