Theorem termcid

Description: The morphism of a terminal category is an identity morphism. (Contributed by Zhi Wang, 16-Oct-2025)

	Ref	Expression
Hypotheses	termcbas.c	⊢ ( 𝜑 → 𝐶 ∈ TermCat )
	termcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	termcbasmo.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	termcbasmo.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	termcid.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	termcid.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
	termcid.i	⊢ 1 = ( Id ‘ 𝐶 )
Assertion	termcid	⊢ ( 𝜑 → 𝐹 = ( 1 ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		termcbas.c	⊢ ( 𝜑 → 𝐶 ∈ TermCat )
2		termcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		termcbasmo.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
4		termcbasmo.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
5		termcid.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
6		termcid.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
7		termcid.i	⊢ 1 = ( Id ‘ 𝐶 )
8	1	termcthind	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
9	1 2 3 4	termcbasmo	⊢ ( 𝜑 → 𝑋 = 𝑌 )
10	9	oveq2d	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑋 ) = ( 𝑋 𝐻 𝑌 ) )
11	6 10	eleqtrrd	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑋 ) )
12	8 2 5 3 7 11	thincid	⊢ ( 𝜑 → 𝐹 = ( 1 ‘ 𝑋 ) )