Theorem sectid

Description: The identity is a section of itself. (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	sectid	|- ( ph -> ( I ` X ) ( X ( Sect ` 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		eqid	|- ( Hom ` C ) = ( Hom ` C )
6		eqid	|- ( comp ` C ) = ( comp ` C )
7	1 5 2 3 4	catidcl	|- ( ph -> ( I ` X ) e. ( X ( Hom ` C ) X ) )
8	1 5 2 3 4 6 4 7	catlid	|- ( ph -> ( ( I ` X ) ( <. X , X >. ( comp ` C ) X ) ( I ` X ) ) = ( I ` X ) )
9		eqid	|- ( Sect ` C ) = ( Sect ` C )
10	1 5 6 2 9 3 4 4 7 7	issect2	|- ( ph -> ( ( I ` X ) ( X ( Sect ` C ) X ) ( I ` X ) <-> ( ( I ` X ) ( <. X , X >. ( comp ` C ) X ) ( I ` X ) ) = ( I ` X ) ) )
11	8 10	mpbird	|- ( ph -> ( I ` X ) ( X ( Sect ` C ) X ) ( I ` X ) )