Theorem invid

Description: The inverse of the identity is the identity. (Contributed by AV, 8-Apr-2020)

	Ref	Expression
Hypotheses	invid.b	|- B = ( Base ` C )
	invid.i	|- I = ( Id ` C )
	invid.c	|- ( ph -> C e. Cat )
	invid.x	|- ( ph -> X e. B )
Assertion	invid	|- ( ph -> ( I ` X ) ( X ( Inv ` C ) X ) ( I ` X ) )

Proof

Step	Hyp	Ref	Expression
1		invid.b	|- B = ( Base ` C )
2		invid.i	|- I = ( Id ` C )
3		invid.c	|- ( ph -> C e. Cat )
4		invid.x	|- ( ph -> X e. B )
5	1 2 3 4	sectid	|- ( ph -> ( I ` X ) ( X ( Sect ` C ) X ) ( I ` X ) )
6		eqid	|- ( Inv ` C ) = ( Inv ` C )
7		eqid	|- ( Sect ` C ) = ( Sect ` C )
8	1 6 3 4 4 7	isinv	|- ( ph -> ( ( I ` X ) ( X ( Inv ` C ) X ) ( I ` X ) <-> ( ( I ` X ) ( X ( Sect ` C ) X ) ( I ` X ) /\ ( I ` X ) ( X ( Sect ` C ) X ) ( I ` X ) ) ) )
9	5 5 8	mpbir2and	|- ( ph -> ( I ` X ) ( X ( Inv ` C ) X ) ( I ` X ) )