Theorem invsym

Description: The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	invfval.b	|- B = ( Base ` C )
	invfval.n	|- N = ( Inv ` C )
	invfval.c	|- ( ph -> C e. Cat )
	invfval.x	|- ( ph -> X e. B )
	invfval.y	|- ( ph -> Y e. B )
Assertion	invsym	|- ( ph -> ( F ( X N Y ) G <-> G ( Y N X ) F ) )

Proof

Step	Hyp	Ref	Expression
1		invfval.b	|- B = ( Base ` C )
2		invfval.n	|- N = ( Inv ` C )
3		invfval.c	|- ( ph -> C e. Cat )
4		invfval.x	|- ( ph -> X e. B )
5		invfval.y	|- ( ph -> Y e. B )
6		eqid	|- ( Sect ` C ) = ( Sect ` C )
7	1 2 3 4 5 6	isinv	|- ( ph -> ( F ( X N Y ) G <-> ( F ( X ( Sect ` C ) Y ) G /\ G ( Y ( Sect ` C ) X ) F ) ) )
8	7	biancomd	|- ( ph -> ( F ( X N Y ) G <-> ( G ( Y ( Sect ` C ) X ) F /\ F ( X ( Sect ` C ) Y ) G ) ) )
9	1 2 3 5 4 6	isinv	|- ( ph -> ( G ( Y N X ) F <-> ( G ( Y ( Sect ` C ) X ) F /\ F ( X ( Sect ` C ) Y ) G ) ) )
10	8 9	bitr4d	|- ( ph -> ( F ( X N Y ) G <-> G ( Y N X ) F ) )