Theorem thinchom

Description: A non-empty hom-set of a thin category is given by its element. (Contributed by Zhi Wang, 20-Oct-2025)

	Ref	Expression
Hypotheses	thinchom.x	|- ( ph -> X e. B )
	thinchom.y	|- ( ph -> Y e. B )
	thinchom.f	|- ( ph -> F e. ( X H Y ) )
	thinchom.b	|- B = ( Base ` C )
	thinchom.h	|- H = ( Hom ` C )
	thinchom.c	|- ( ph -> C e. ThinCat )
Assertion	thinchom	|- ( ph -> ( X H Y ) = { F } )

Proof

Step	Hyp	Ref	Expression
1		thinchom.x	|- ( ph -> X e. B )
2		thinchom.y	|- ( ph -> Y e. B )
3		thinchom.f	|- ( ph -> F e. ( X H Y ) )
4		thinchom.b	|- B = ( Base ` C )
5		thinchom.h	|- H = ( Hom ` C )
6		thinchom.c	|- ( ph -> C e. ThinCat )
7	1	adantr	|- ( ( ph /\ g e. ( X H Y ) ) -> X e. B )
8	2	adantr	|- ( ( ph /\ g e. ( X H Y ) ) -> Y e. B )
9		simpr	|- ( ( ph /\ g e. ( X H Y ) ) -> g e. ( X H Y ) )
10	3	adantr	|- ( ( ph /\ g e. ( X H Y ) ) -> F e. ( X H Y ) )
11	6	adantr	|- ( ( ph /\ g e. ( X H Y ) ) -> C e. ThinCat )
12	7 8 9 10 4 5 11	thincmo2	|- ( ( ph /\ g e. ( X H Y ) ) -> g = F )
13	12 3	eqsnd	|- ( ph -> ( X H Y ) = { F } )