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	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	thinchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	thinchom.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
	thinchom.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	thinchom.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	thinchom.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
Assertion	thinchom	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = { 𝐹 } )

Proof

Step	Hyp	Ref	Expression
1		thinchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
2		thinchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
3		thinchom.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
4		thinchom.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
5		thinchom.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
6		thinchom.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
7	1	adantr	⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝑋 ∈ 𝐵 )
8	2	adantr	⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝑌 ∈ 𝐵 )
9		simpr	⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) )
10	3	adantr	⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
11	6	adantr	⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝐶 ∈ ThinCat )
12	7 8 9 10 4 5 11	thincmo2	⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝑔 = 𝐹 )
13	12 3	eqsnd	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = { 𝐹 } )