Theorem gicen

Description: Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015)

	Ref	Expression
Hypotheses	gicen.b	|- B = ( Base ` R )
	gicen.c	|- C = ( Base ` S )
Assertion	gicen	|- ( R ~=g S -> B ~~ C )

Proof

Step	Hyp	Ref	Expression
1		gicen.b	|- B = ( Base ` R )
2		gicen.c	|- C = ( Base ` S )
3		brgic	|- ( R ~=g S <-> ( R GrpIso S ) =/= (/) )
4		n0	|- ( ( R GrpIso S ) =/= (/) <-> E. f f e. ( R GrpIso S ) )
5	1 2	gimf1o	|- ( f e. ( R GrpIso S ) -> f : B -1-1-onto-> C )
6	1	fvexi	|- B e. _V
7	6	f1oen	|- ( f : B -1-1-onto-> C -> B ~~ C )
8	5 7	syl	|- ( f e. ( R GrpIso S ) -> B ~~ C )
9	8	exlimiv	|- ( E. f f e. ( R GrpIso S ) -> B ~~ C )
10	4 9	sylbi	|- ( ( R GrpIso S ) =/= (/) -> B ~~ C )
11	3 10	sylbi	|- ( R ~=g S -> B ~~ C )