Theorem hmphi

Description: If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015)

	Ref	Expression
Assertion	hmphi	⊢ ( 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) → 𝐽 ≃ 𝐾 )

Proof

Step	Hyp	Ref	Expression
1		ne0i	⊢ ( 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) → ( 𝐽 Homeo 𝐾 ) ≠ ∅ )
2		hmph	⊢ ( 𝐽 ≃ 𝐾 ↔ ( 𝐽 Homeo 𝐾 ) ≠ ∅ )
3	1 2	sylibr	⊢ ( 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) → 𝐽 ≃ 𝐾 )