Theorem termcid2

Description: The morphism of a terminal category is an identity morphism. (Contributed by Zhi Wang, 16-Oct-2025)

	Ref	Expression
Hypotheses	termcbas.c	|- ( ph -> C e. TermCat )
	termcbas.b	|- B = ( Base ` C )
	termcbasmo.x	|- ( ph -> X e. B )
	termcbasmo.y	|- ( ph -> Y e. B )
	termcid.h	|- H = ( Hom ` C )
	termcid.f	|- ( ph -> F e. ( X H Y ) )
	termcid.i	|- .1. = ( Id ` C )
Assertion	termcid2	|- ( ph -> F = ( .1. ` Y ) )

Proof

Step	Hyp	Ref	Expression
1		termcbas.c	|- ( ph -> C e. TermCat )
2		termcbas.b	|- B = ( Base ` C )
3		termcbasmo.x	|- ( ph -> X e. B )
4		termcbasmo.y	|- ( ph -> Y e. B )
5		termcid.h	|- H = ( Hom ` C )
6		termcid.f	|- ( ph -> F e. ( X H Y ) )
7		termcid.i	|- .1. = ( Id ` C )
8	1 2 3 4 5 6 7	termcid	|- ( ph -> F = ( .1. ` X ) )
9	1 2 3 4	termcbasmo	|- ( ph -> X = Y )
10	9	fveq2d	|- ( ph -> ( .1. ` X ) = ( .1. ` Y ) )
11	8 10	eqtrd	|- ( ph -> F = ( .1. ` Y ) )