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Metamath Proof Explorer

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	$⊢ F \in (J Homeo K) \to J ≃ K$

Step	Hyp	Ref	Expression
1		ne0i	$⊢ F \in (J Homeo K) \to J Homeo K \neq \emptyset$
2		hmph	$⊢ J ≃ K \leftrightarrow J Homeo K \neq \emptyset$
3	1 2	sylibr	$⊢ F \in (J Homeo K) \to J ≃ K$