Theorem initorcl

Description: Reverse closure for an initial object: If a class has an initial object, the class is a category. (Contributed by AV, 4-Apr-2020)

	Ref	Expression
Assertion	initorcl	$$⊢I\in \mathrm{InitO}\left(C\right)\to C\in \mathrm{Cat}$$

Proof

Step	Hyp	Ref	Expression
1		df-inito	$$⊢\mathrm{InitO}=\left(c\in \mathrm{Cat}⟼\left\{a\in {\mathrm{Base}}_c|\forall b\in {\mathrm{Base}}_c\exists !hh\in \left(a\mathrm{Hom}\left(c\right)b\right)\right\}\right)$$
2	1	mptrcl	$$⊢I\in \mathrm{InitO}\left(C\right)\to C\in \mathrm{Cat}$$