Theorem termchom

Description: The hom-set of a terminal category is a singleton of the identity morphism. (Contributed by Zhi Wang, 20-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 )
Assertion	termchom	|- ( ph -> ( X H Y ) = { ( .1. ` X ) } )

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	1 2 3 4 5	termchomn0	|- ( ph -> -. ( X H Y ) = (/) )
8		neq0	|- ( -. ( X H Y ) = (/) <-> E. f f e. ( X H Y ) )
9	7 8	sylib	|- ( ph -> E. f f e. ( X H Y ) )
10	3	adantr	|- ( ( ph /\ f e. ( X H Y ) ) -> X e. B )
11	4	adantr	|- ( ( ph /\ f e. ( X H Y ) ) -> Y e. B )
12		simpr	|- ( ( ph /\ f e. ( X H Y ) ) -> f e. ( X H Y ) )
13	1	adantr	|- ( ( ph /\ f e. ( X H Y ) ) -> C e. TermCat )
14	13	termcthind	|- ( ( ph /\ f e. ( X H Y ) ) -> C e. ThinCat )
15	10 11 12 2 5 14	thinchom	|- ( ( ph /\ f e. ( X H Y ) ) -> ( X H Y ) = { f } )
16	13 2 10 11 5 12 6	termcid	|- ( ( ph /\ f e. ( X H Y ) ) -> f = ( .1. ` X ) )
17	16	sneqd	|- ( ( ph /\ f e. ( X H Y ) ) -> { f } = { ( .1. ` X ) } )
18	15 17	eqtrd	|- ( ( ph /\ f e. ( X H Y ) ) -> ( X H Y ) = { ( .1. ` X ) } )
19	9 18	exlimddv	|- ( ph -> ( X H Y ) = { ( .1. ` X ) } )