Theorem gimf1o

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

	Ref	Expression
Hypotheses	isgim.b	|- B = ( Base ` R )
	isgim.c	|- C = ( Base ` S )
Assertion	gimf1o	|- ( F e. ( R GrpIso S ) -> F : B -1-1-onto-> C )

Proof

Step	Hyp	Ref	Expression
1		isgim.b	|- B = ( Base ` R )
2		isgim.c	|- C = ( Base ` S )
3	1 2	isgim	|- ( F e. ( R GrpIso S ) <-> ( F e. ( R GrpHom S ) /\ F : B -1-1-onto-> C ) )
4	3	simprbi	|- ( F e. ( R GrpIso S ) -> F : B -1-1-onto-> C )