Theorem isinito2

Description: The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 23-Oct-2025)

	Ref	Expression
Hypotheses	isinito2.1	|- .1. = ( SetCat ` 1o )
	isinito2.f	|- F = ( ( 1st ` ( .1. DiagFunc C ) ) ` (/) )
Assertion	isinito2	|- ( I e. ( InitO ` C ) <-> I ( F ( C UP .1. ) (/) ) (/) )

Proof

Step	Hyp	Ref	Expression
1		isinito2.1	|- .1. = ( SetCat ` 1o )
2		isinito2.f	|- F = ( ( 1st ` ( .1. DiagFunc C ) ) ` (/) )
3		initorcl	|- ( I e. ( InitO ` C ) -> C e. Cat )
4		eqid	|- ( Base ` C ) = ( Base ` C )
5	4	initoo2	|- ( I e. ( InitO ` C ) -> I e. ( Base ` C ) )
6	1 2 3 5	isinito2lem	|- ( I e. ( InitO ` C ) -> ( I e. ( InitO ` C ) <-> I ( F ( C UP .1. ) (/) ) (/) ) )
7	6	ibi	|- ( I e. ( InitO ` C ) -> I ( F ( C UP .1. ) (/) ) (/) )
8		id	|- ( I ( F ( C UP .1. ) (/) ) (/) -> I ( F ( C UP .1. ) (/) ) (/) )
9	8	up1st2nd	|- ( I ( F ( C UP .1. ) (/) ) (/) -> I ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( C UP .1. ) (/) ) (/) )
10	9	uprcl2	|- ( I ( F ( C UP .1. ) (/) ) (/) -> ( 1st ` F ) ( C Func .1. ) ( 2nd ` F ) )
11	10	funcrcl2	|- ( I ( F ( C UP .1. ) (/) ) (/) -> C e. Cat )
12	9 4	uprcl4	|- ( I ( F ( C UP .1. ) (/) ) (/) -> I e. ( Base ` C ) )
13	1 2 11 12	isinito2lem	|- ( I ( F ( C UP .1. ) (/) ) (/) -> ( I e. ( InitO ` C ) <-> I ( F ( C UP .1. ) (/) ) (/) ) )
14	13	ibir	|- ( I ( F ( C UP .1. ) (/) ) (/) -> I e. ( InitO ` C ) )
15	7 14	impbii	|- ( I e. ( InitO ` C ) <-> I ( F ( C UP .1. ) (/) ) (/) )