Theorem termcid2

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	termcid2	$$⊢\phi \to F=\underset{\dot{}}{1}\left(Y\right)$$

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 2 3 4 5 6 7	termcid	$$⊢\phi \to F=\underset{\dot{}}{1}\left(X\right)$$
9	1 2 3 4	termcbasmo	$$⊢\phi \to X=Y$$
10	9	fveq2d	$$⊢\phi \to \underset{\dot{}}{1}\left(X\right)=\underset{\dot{}}{1}\left(Y\right)$$
11	8 10	eqtrd	$$⊢\phi \to F=\underset{\dot{}}{1}\left(Y\right)$$