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	$$⊢F\in \left(J\mathrm{Homeo}K\right)\to J\simeq K$$

Proof

Step	Hyp	Ref	Expression
1		ne0i	$$⊢F\in \left(J\mathrm{Homeo}K\right)\to J\mathrm{Homeo}K\ne \varnothing$$
2		hmph	$$⊢J\simeq K\leftrightarrow J\mathrm{Homeo}K\ne \varnothing$$
3	1 2	sylibr	$$⊢F\in \left(J\mathrm{Homeo}K\right)\to J\simeq K$$