Theorem ishmeo

Description: The predicate F is a homeomorphism between topology J and topology K . Criterion of BourbakiTop1 p. I.2. (Contributed by FL, 14-Feb-2007) (Revised by Mario Carneiro, 22-Aug-2015)

	Ref	Expression
Assertion	ishmeo	$$⊢F\in \left(J\mathrm{Homeo}K\right)\leftrightarrow \left(F\in \left(J\mathrm{Cn}K\right)\wedge F^{-1}\in \left(K\mathrm{Cn}J\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cnveq	$$⊢f=F\to f^{-1}=F^{-1}$$
2	1	eleq1d	$$⊢f=F\to \left(f^{-1}\in \left(K\mathrm{Cn}J\right)\leftrightarrow F^{-1}\in \left(K\mathrm{Cn}J\right)\right)$$
3		hmeofval	$$⊢J\mathrm{Homeo}K=\left\{f\in \left(J\mathrm{Cn}K\right)|f^{-1}\in \left(K\mathrm{Cn}J\right)\right\}$$
4	2 3	elrab2	$$⊢F\in \left(J\mathrm{Homeo}K\right)\leftrightarrow \left(F\in \left(J\mathrm{Cn}K\right)\wedge F^{-1}\in \left(K\mathrm{Cn}J\right)\right)$$