Theorem termchom2

Description: The hom-set of a terminal category is a singleton of the identity morphism. (Contributed by Zhi Wang, 21-Oct-2025)

	Ref	Expression
Hypotheses	termchom.c	|- ( ph -> C e. TermCat )
	termchom.b	|- B = ( Base ` C )
	termchom.x	|- ( ph -> X e. B )
	termchom.y	|- ( ph -> Y e. B )
	termchom.h	|- H = ( Hom ` C )
	termchom.i	|- .1. = ( Id ` C )
	termchom2.z	|- ( ph -> Z e. B )
Assertion	termchom2	|- ( ph -> ( X H Y ) = { ( .1. ` Z ) } )

Proof

Step	Hyp	Ref	Expression
1		termchom.c	|- ( ph -> C e. TermCat )
2		termchom.b	|- B = ( Base ` C )
3		termchom.x	|- ( ph -> X e. B )
4		termchom.y	|- ( ph -> Y e. B )
5		termchom.h	|- H = ( Hom ` C )
6		termchom.i	|- .1. = ( Id ` C )
7		termchom2.z	|- ( ph -> Z e. B )
8	1 2 3 4 5 6	termchom	|- ( ph -> ( X H Y ) = { ( .1. ` X ) } )
9	1 2 3 7	termcbasmo	|- ( ph -> X = Z )
10	9	fveq2d	|- ( ph -> ( .1. ` X ) = ( .1. ` Z ) )
11	10	sneqd	|- ( ph -> { ( .1. ` X ) } = { ( .1. ` Z ) } )
12	8 11	eqtrd	|- ( ph -> ( X H Y ) = { ( .1. ` Z ) } )