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	|- ( ph -> C e. TermCat )
	termcbas.b	|- B = ( Base ` C )
	termcbasmo.x	|- ( ph -> X e. B )
	termcbasmo.y	|- ( ph -> Y e. B )
	termcid.h	|- H = ( Hom ` C )
	termcid.f	|- ( ph -> F e. ( X H Y ) )
	termcid.i	|- .1. = ( Id ` C )
Assertion	termcid	|- ( ph -> F = ( .1. ` X ) )

Proof

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