This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) )