Theorem funcsetc1ocl

Description: The functor to the trivial category. The converse is also true due to reverse closure. (Contributed by Zhi Wang, 22-Oct-2025)

	Ref	Expression
Hypotheses	funcsetc1o.1	|- .1. = ( SetCat ` 1o )
	funcsetc1o.f	|- F = ( ( 1st ` ( .1. DiagFunc C ) ) ` (/) )
	funcsetc1o.c	|- ( ph -> C e. Cat )
Assertion	funcsetc1ocl	|- ( ph -> F e. ( C Func .1. ) )

Proof

Step	Hyp	Ref	Expression
1		funcsetc1o.1	|- .1. = ( SetCat ` 1o )
2		funcsetc1o.f	|- F = ( ( 1st ` ( .1. DiagFunc C ) ) ` (/) )
3		funcsetc1o.c	|- ( ph -> C e. Cat )
4		eqid	|- ( .1. DiagFunc C ) = ( .1. DiagFunc C )
5		setc1oterm	|- ( SetCat ` 1o ) e. TermCat
6	1 5	eqeltri	|- .1. e. TermCat
7	6	a1i	|- ( ph -> .1. e. TermCat )
8	7	termccd	|- ( ph -> .1. e. Cat )
9	1	setc1obas	|- 1o = ( Base ` .1. )
10		0lt1o	|- (/) e. 1o
11	10	a1i	|- ( ph -> (/) e. 1o )
12	4 8 3 9 11 2	diag1cl	|- ( ph -> F e. ( C Func .1. ) )