Theorem idinv

Description: The inverse of the identity is the identity. Example 3.13 of Adamek p. 28. (Contributed by AV, 9-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	idinv	|- ( ph -> ( ( X ( Inv ` C ) X ) ` ( I ` 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	|- ( Inv ` C ) = ( Inv ` C )
6	1 5 3 4 4	invfun	|- ( ph -> Fun ( X ( Inv ` C ) X ) )
7	1 2 3 4	invid	|- ( ph -> ( I ` X ) ( X ( Inv ` C ) X ) ( I ` X ) )
8		funbrfv	|- ( Fun ( X ( Inv ` C ) X ) -> ( ( I ` X ) ( X ( Inv ` C ) X ) ( I ` X ) -> ( ( X ( Inv ` C ) X ) ` ( I ` X ) ) = ( I ` X ) ) )
9	6 7 8	sylc	|- ( ph -> ( ( X ( Inv ` C ) X ) ` ( I ` X ) ) = ( I ` X ) )