Theorem thinccat

Description: A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024)

	Ref	Expression
Assertion	thinccat	$$⊢C\in \mathrm{ThinCat}\to C\in \mathrm{Cat}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_C={\mathrm{Base}}_C$$
2		eqid	$$⊢\mathrm{Hom}\left(C\right)=\mathrm{Hom}\left(C\right)$$
3	1 2	isthinc	$$⊢C\in \mathrm{ThinCat}\leftrightarrow \left(C\in \mathrm{Cat}\wedge \forall x\in {\mathrm{Base}}_C\forall y\in {\mathrm{Base}}_C{\exists }^{*}ff\in \left(x\mathrm{Hom}\left(C\right)y\right)\right)$$
4	3	simplbi	$$⊢C\in \mathrm{ThinCat}\to C\in \mathrm{Cat}$$