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Metamath Proof Explorer

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	$⊢ \underset{˙}{1} = SetCat (1_{𝑜})$
		isinito2.f	$⊢ F = 1^{st} (\underset{˙}{1} Δ_{func} C) (\emptyset)$
	Assertion	isinito2	Could not format assertion : No typesetting found for \|- ( I e. ( InitO ` C ) <-> I ( F ( C UP .1. ) (/) ) (/) ) with typecode \|-

Proof

Step	Hyp	Ref	Expression
1		isinito2.1	$⊢ \underset{˙}{1} = SetCat (1_{𝑜})$
2		isinito2.f	$⊢ F = 1^{st} (\underset{˙}{1} Δ_{func} C) (\emptyset)$
3		initorcl	$⊢ I \in InitO (C) \to C \in Cat$
4		eqid	$⊢ {Base}_{C} = {Base}_{C}$
5	4	initoo2	$⊢ I \in InitO (C) \to I \in {Base}_{C}$
6	1 2 3 5	isinito2lem	Could not format ( I e. ( InitO ` C ) -> ( I e. ( InitO ` C ) <-> I ( F ( C UP .1. ) (/) ) (/) ) ) : No typesetting found for \|- ( I e. ( InitO ` C ) -> ( I e. ( InitO ` C ) <-> I ( F ( C UP .1. ) (/) ) (/) ) ) with typecode \|-
7	6	ibi	Could not format ( I e. ( InitO ` C ) -> I ( F ( C UP .1. ) (/) ) (/) ) : No typesetting found for \|- ( I e. ( InitO ` C ) -> I ( F ( C UP .1. ) (/) ) (/) ) with typecode \|-
8		id	Could not format ( I ( F ( C UP .1. ) (/) ) (/) -> I ( F ( C UP .1. ) (/) ) (/) ) : No typesetting found for \|- ( I ( F ( C UP .1. ) (/) ) (/) -> I ( F ( C UP .1. ) (/) ) (/) ) with typecode \|-
9	8	up1st2nd	Could not format ( I ( F ( C UP .1. ) (/) ) (/) -> I ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( C UP .1. ) (/) ) (/) ) : No typesetting found for \|- ( I ( F ( C UP .1. ) (/) ) (/) -> I ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( C UP .1. ) (/) ) (/) ) with typecode \|-
10	9	uprcl2	Could not format ( I ( F ( C UP .1. ) (/) ) (/) -> ( 1st ` F ) ( C Func .1. ) ( 2nd ` F ) ) : No typesetting found for \|- ( I ( F ( C UP .1. ) (/) ) (/) -> ( 1st ` F ) ( C Func .1. ) ( 2nd ` F ) ) with typecode \|-
11	10	funcrcl2	Could not format ( I ( F ( C UP .1. ) (/) ) (/) -> C e. Cat ) : No typesetting found for \|- ( I ( F ( C UP .1. ) (/) ) (/) -> C e. Cat ) with typecode \|-
12	9 4	uprcl4	Could not format ( I ( F ( C UP .1. ) (/) ) (/) -> I e. ( Base ` C ) ) : No typesetting found for \|- ( I ( F ( C UP .1. ) (/) ) (/) -> I e. ( Base ` C ) ) with typecode \|-
13	1 2 11 12	isinito2lem	Could not format ( I ( F ( C UP .1. ) (/) ) (/) -> ( I e. ( InitO ` C ) <-> I ( F ( C UP .1. ) (/) ) (/) ) ) : No typesetting found for \|- ( I ( F ( C UP .1. ) (/) ) (/) -> ( I e. ( InitO ` C ) <-> I ( F ( C UP .1. ) (/) ) (/) ) ) with typecode \|-
14	13	ibir	Could not format ( I ( F ( C UP .1. ) (/) ) (/) -> I e. ( InitO ` C ) ) : No typesetting found for \|- ( I ( F ( C UP .1. ) (/) ) (/) -> I e. ( InitO ` C ) ) with typecode \|-
15	7 14	impbii	Could not format ( I e. ( InitO ` C ) <-> I ( F ( C UP .1. ) (/) ) (/) ) : No typesetting found for \|- ( I e. ( InitO ` C ) <-> I ( F ( C UP .1. ) (/) ) (/) ) with typecode \|-