Theorem thincid

Description: In a thin category, a morphism from an object to itself is an identity morphism. (Contributed by Zhi Wang, 24-Sep-2024)

	Ref	Expression
Hypotheses	thincid.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
	thincid.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	thincid.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	thincid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	thincid.i	⊢ 1 = ( Id ‘ 𝐶 )
	thincid.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑋 ) )
Assertion	thincid	⊢ ( 𝜑 → 𝐹 = ( 1 ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		thincid.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
2		thincid.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		thincid.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
4		thincid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		thincid.i	⊢ 1 = ( Id ‘ 𝐶 )
6		thincid.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑋 ) )
7	1	thinccatd	⊢ ( 𝜑 → 𝐶 ∈ Cat )
8	2 3 5 7 4	catidcl	⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) ∈ ( 𝑋 𝐻 𝑋 ) )
9	4 4 6 8 2 3 1	thincmo2	⊢ ( 𝜑 → 𝐹 = ( 1 ‘ 𝑋 ) )