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	|- ( ph -> C e. ThinCat )
	thincid.b	|- B = ( Base ` C )
	thincid.h	|- H = ( Hom ` C )
	thincid.x	|- ( ph -> X e. B )
	thincid.i	|- .1. = ( Id ` C )
	thincid.f	|- ( ph -> F e. ( X H X ) )
Assertion	thincid	|- ( ph -> F = ( .1. ` X ) )

Proof

Step	Hyp	Ref	Expression
1		thincid.c	|- ( ph -> C e. ThinCat )
2		thincid.b	|- B = ( Base ` C )
3		thincid.h	|- H = ( Hom ` C )
4		thincid.x	|- ( ph -> X e. B )
5		thincid.i	|- .1. = ( Id ` C )
6		thincid.f	|- ( ph -> F e. ( X H X ) )
7	1	thinccd	|- ( ph -> C e. Cat )
8	2 3 5 7 4	catidcl	|- ( ph -> ( .1. ` X ) e. ( X H X ) )
9	4 4 6 8 2 3 1	thincmo2	|- ( ph -> F = ( .1. ` X ) )