Theorem prstcbas

Description: The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024)

	Ref	Expression
Hypotheses	prstcnid.c	$$⊢\phi \to C=\mathrm{ProsetToCat}\left(K\right)$$
	prstcnid.k	$$⊢\phi \to K\in \mathrm{Proset}$$
	prstcbas.b	$$⊢\phi \to B={\mathrm{Base}}_K$$
Assertion	prstcbas	$$⊢\phi \to B={\mathrm{Base}}_C$$

Proof

Step	Hyp	Ref	Expression
1		prstcnid.c	$$⊢\phi \to C=\mathrm{ProsetToCat}\left(K\right)$$
2		prstcnid.k	$$⊢\phi \to K\in \mathrm{Proset}$$
3		prstcbas.b	$$⊢\phi \to B={\mathrm{Base}}_K$$
4		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
5		slotsbhcdif	$$⊢\left({\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{Hom}\left(\mathrm{ndx}\right)\wedge {\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{comp}\left(\mathrm{ndx}\right)\wedge \mathrm{Hom}\left(\mathrm{ndx}\right)\ne \mathrm{comp}\left(\mathrm{ndx}\right)\right)$$
6	5	simp2i	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{comp}\left(\mathrm{ndx}\right)$$
7	5	simp1i	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{Hom}\left(\mathrm{ndx}\right)$$
8	1 2 4 6 7	prstcnid	$$⊢\phi \to {\mathrm{Base}}_K={\mathrm{Base}}_C$$
9	3 8	eqtrd	$$⊢\phi \to B={\mathrm{Base}}_C$$