Theorem mndtcid

Description: The identity morphism, or identity arrow, of the category built from a monoid is the identity element of the monoid. (Contributed by Zhi Wang, 22-Sep-2024)

	Ref	Expression
Hypotheses	mndtccat.c	|- ( ph -> C = ( MndToCat ` M ) )
	mndtccat.m	|- ( ph -> M e. Mnd )
	mndtcid.b	|- ( ph -> B = ( Base ` C ) )
	mndtcid.x	|- ( ph -> X e. B )
	mndtcid.i	|- ( ph -> .1. = ( Id ` C ) )
Assertion	mndtcid	|- ( ph -> ( .1. ` X ) = ( 0g ` M ) )

Proof

Step	Hyp	Ref	Expression
1		mndtccat.c	|- ( ph -> C = ( MndToCat ` M ) )
2		mndtccat.m	|- ( ph -> M e. Mnd )
3		mndtcid.b	|- ( ph -> B = ( Base ` C ) )
4		mndtcid.x	|- ( ph -> X e. B )
5		mndtcid.i	|- ( ph -> .1. = ( Id ` C ) )
6	1 2	mndtccatid	|- ( ph -> ( C e. Cat /\ ( Id ` C ) = ( x e. ( Base ` C ) |-> ( 0g ` M ) ) ) )
7	6	simprd	|- ( ph -> ( Id ` C ) = ( x e. ( Base ` C ) |-> ( 0g ` M ) ) )
8	5 7	eqtrd	|- ( ph -> .1. = ( x e. ( Base ` C ) |-> ( 0g ` M ) ) )
9		eqidd	|- ( ( ph /\ x = X ) -> ( 0g ` M ) = ( 0g ` M ) )
10	4 3	eleqtrd	|- ( ph -> X e. ( Base ` C ) )
11		fvexd	|- ( ph -> ( 0g ` M ) e. _V )
12	8 9 10 11	fvmptd	|- ( ph -> ( .1. ` X ) = ( 0g ` M ) )