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	No typesetting found for |- ( ph -> C e. TermCat ) with typecode |-
	termcbas.b	$$⊢B={\mathrm{Base}}_C$$
	termcbasmo.x	$$⊢\phi \to X\in B$$
	termcbasmo.y	$$⊢\phi \to Y\in B$$
	termcid.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
	termcid.f	$$⊢\phi \to F\in \left(XHY\right)$$
	termcid.i	$$⊢\underset{\dot{}}{1}=\mathrm{Id}\left(C\right)$$
Assertion	termcid	$$⊢\phi \to F=\underset{\dot{}}{1}\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		termcbas.c	Could not format ( ph -> C e. TermCat ) : No typesetting found for |- ( ph -> C e. TermCat ) with typecode |-
2		termcbas.b	$$⊢B={\mathrm{Base}}_C$$
3		termcbasmo.x	$$⊢\phi \to X\in B$$
4		termcbasmo.y	$$⊢\phi \to Y\in B$$
5		termcid.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
6		termcid.f	$$⊢\phi \to F\in \left(XHY\right)$$
7		termcid.i	$$⊢\underset{\dot{}}{1}=\mathrm{Id}\left(C\right)$$
8	1	termcthind	$$⊢\phi \to C\in \mathrm{ThinCat}$$
9	1 2 3 4	termcbasmo	$$⊢\phi \to X=Y$$
10	9	oveq2d	$$⊢\phi \to XHX=XHY$$
11	6 10	eleqtrrd	$$⊢\phi \to F\in \left(XHX\right)$$
12	8 2 5 3 7 11	thincid	$$⊢\phi \to F=\underset{\dot{}}{1}\left(X\right)$$