Theorem estrccat

Description: The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020)

	Ref	Expression
Hypothesis	estrccat.c	$$⊢C=\mathrm{ExtStrCat}\left(U\right)$$
Assertion	estrccat	$$⊢U\in V\to C\in \mathrm{Cat}$$

Proof

Step	Hyp	Ref	Expression
1		estrccat.c	$$⊢C=\mathrm{ExtStrCat}\left(U\right)$$
2	1	estrccatid	$$⊢U\in V\to \left(C\in \mathrm{Cat}\wedge \mathrm{Id}\left(C\right)=\left(x\in U⟼{\mathrm{I}\upharpoonright }_{{\mathrm{Base}}_x}\right)\right)$$
3	2	simpld	$$⊢U\in V\to C\in \mathrm{Cat}$$