Theorem isgim2

Description: A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo . (Contributed by Mario Carneiro, 6-May-2015)

	Ref	Expression
Assertion	isgim2	|- ( F e. ( R GrpIso S ) <-> ( F e. ( R GrpHom S ) /\ `' F e. ( S GrpHom R ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( Base ` R ) = ( Base ` R )
2		eqid	|- ( Base ` S ) = ( Base ` S )
3	1 2	isgim	|- ( F e. ( R GrpIso S ) <-> ( F e. ( R GrpHom S ) /\ F : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) )
4	1 2	ghmf1o	|- ( F e. ( R GrpHom S ) -> ( F : ( Base ` R ) -1-1-onto-> ( Base ` S ) <-> `' F e. ( S GrpHom R ) ) )
5	4	pm5.32i	|- ( ( F e. ( R GrpHom S ) /\ F : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) <-> ( F e. ( R GrpHom S ) /\ `' F e. ( S GrpHom R ) ) )
6	3 5	bitri	|- ( F e. ( R GrpIso S ) <-> ( F e. ( R GrpHom S ) /\ `' F e. ( S GrpHom R ) ) )