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 e. ( J Homeo K ) <-> ( F e. ( J Cn K ) /\ `' F e. ( K Cn J ) ) )

Proof

Step	Hyp	Ref	Expression
1		cnveq	|- ( f = F -> `' f = `' F )
2	1	eleq1d	|- ( f = F -> ( `' f e. ( K Cn J ) <-> `' F e. ( K Cn J ) ) )
3		hmeofval	|- ( J Homeo K ) = { f e. ( J Cn K ) | `' f e. ( K Cn J ) }
4	2 3	elrab2	|- ( F e. ( J Homeo K ) <-> ( F e. ( J Cn K ) /\ `' F e. ( K Cn J ) ) )