Theorem gicsym

Description: Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Assertion	gicsym	|- ( R ~=g S -> S ~=g R )

Proof

Step	Hyp	Ref	Expression
1		brgic	|- ( R ~=g S <-> ( R GrpIso S ) =/= (/) )
2		n0	|- ( ( R GrpIso S ) =/= (/) <-> E. f f e. ( R GrpIso S ) )
3		gimcnv	|- ( f e. ( R GrpIso S ) -> `' f e. ( S GrpIso R ) )
4		brgici	|- ( `' f e. ( S GrpIso R ) -> S ~=g R )
5	3 4	syl	|- ( f e. ( R GrpIso S ) -> S ~=g R )
6	5	exlimiv	|- ( E. f f e. ( R GrpIso S ) -> S ~=g R )
7	2 6	sylbi	|- ( ( R GrpIso S ) =/= (/) -> S ~=g R )
8	1 7	sylbi	|- ( R ~=g S -> S ~=g R )