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	⊢ ( 𝜑 → 𝐶 = ( MndToCat ‘ 𝑀 ) )
	mndtccat.m	⊢ ( 𝜑 → 𝑀 ∈ Mnd )
	mndtcid.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
	mndtcid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	mndtcid.i	⊢ ( 𝜑 → 1 = ( Id ‘ 𝐶 ) )
Assertion	mndtcid	⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) = ( 0g ‘ 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		mndtccat.c	⊢ ( 𝜑 → 𝐶 = ( MndToCat ‘ 𝑀 ) )
2		mndtccat.m	⊢ ( 𝜑 → 𝑀 ∈ Mnd )
3		mndtcid.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
4		mndtcid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		mndtcid.i	⊢ ( 𝜑 → 1 = ( Id ‘ 𝐶 ) )
6	1 2	mndtccatid	⊢ ( 𝜑 → ( 𝐶 ∈ Cat ∧ ( Id ‘ 𝐶 ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( 0g ‘ 𝑀 ) ) ) )
7	6	simprd	⊢ ( 𝜑 → ( Id ‘ 𝐶 ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( 0g ‘ 𝑀 ) ) )
8	5 7	eqtrd	⊢ ( 𝜑 → 1 = ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( 0g ‘ 𝑀 ) ) )
9		eqidd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 ) )
10	4 3	eleqtrd	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
11		fvexd	⊢ ( 𝜑 → ( 0g ‘ 𝑀 ) ∈ V )
12	8 9 10 11	fvmptd	⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) = ( 0g ‘ 𝑀 ) )