Theorem thincmod

Description: At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024)

	Ref	Expression
Hypotheses	thincmo.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
	thincmo.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	thincmo.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	thincn0eu.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
	thincn0eu.h	⊢ ( 𝜑 → 𝐻 = ( Hom ‘ 𝐶 ) )
Assertion	thincmod	⊢ ( 𝜑 → ∃* 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		thincmo.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
2		thincmo.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
3		thincmo.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
4		thincn0eu.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
5		thincn0eu.h	⊢ ( 𝜑 → 𝐻 = ( Hom ‘ 𝐶 ) )
6	2 4	eleqtrd	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
7	3 4	eleqtrd	⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐶 ) )
8		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
9		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
10	1 6 7 8 9	thincmo	⊢ ( 𝜑 → ∃* 𝑓 𝑓 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
11	5	oveqd	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
12	11	eleq2d	⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝑓 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) )
13	12	mobidv	⊢ ( 𝜑 → ( ∃* 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↔ ∃* 𝑓 𝑓 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) )
14	10 13	mpbird	⊢ ( 𝜑 → ∃* 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) )