Theorem setc1oid

Description: The identity morphism of the trivial category. (Contributed by Zhi Wang, 22-Oct-2025)

	Ref	Expression
Hypotheses	funcsetc1o.1	|- .1. = ( SetCat ` 1o )
	setc1oid.i	|- I = ( Id ` .1. )
Assertion	setc1oid	|- ( I ` (/) ) = (/)

Proof

Step	Hyp	Ref	Expression
1		funcsetc1o.1	|- .1. = ( SetCat ` 1o )
2		setc1oid.i	|- I = ( Id ` .1. )
3		1oex	|- 1o e. _V
4	3	a1i	|- ( T. -> 1o e. _V )
5		0lt1o	|- (/) e. 1o
6	5	a1i	|- ( T. -> (/) e. 1o )
7	1 2 4 6	setcid	|- ( T. -> ( I ` (/) ) = ( _I |` (/) ) )
8	7	mptru	|- ( I ` (/) ) = ( _I |` (/) )
9		res0	|- ( _I |` (/) ) = (/)
10	8 9	eqtri	|- ( I ` (/) ) = (/)