Metamath Proof Explorer

Theorem gimghm

Description: An isomorphism of groups is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015) (Revised by Mario Carneiro, 6-May-2015)

Ref Expression
Assertion gimghm 
|- ( F e. ( R GrpIso S ) -> F e. ( R GrpHom S ) )

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 3 simplbi 
 |-  ( F e. ( R GrpIso S ) -> F e. ( R GrpHom S ) )