Theorem initofn

Description: InitO is a function on Cat . (Contributed by Zhi Wang, 29-Aug-2024)

	Ref	Expression
Assertion	initofn	$$⊢\mathrm{InitO}Fn\mathrm{Cat}$$

Proof

Step	Hyp	Ref	Expression
1		fvex	$$⊢{\mathrm{Base}}_c\in \mathrm{V}$$
2	1	rabex	$$⊢\left\{a\in {\mathrm{Base}}_c|\forall b\in {\mathrm{Base}}_c\exists !hh\in \left(a\mathrm{Hom}\left(c\right)b\right)\right\}\in \mathrm{V}$$
3		df-inito	$$⊢\mathrm{InitO}=\left(c\in \mathrm{Cat}⟼\left\{a\in {\mathrm{Base}}_c|\forall b\in {\mathrm{Base}}_c\exists !hh\in \left(a\mathrm{Hom}\left(c\right)b\right)\right\}\right)$$
4	2 3	fnmpti	$$⊢\mathrm{InitO}Fn\mathrm{Cat}$$