Lemma PreFilterBase_Generated {X : UU} (base : BaseOfPreFilter X) : PreFilterBase base = PreFilterGenerated base. Proof. simple refine (subtypePath_prop (B := λ _, make_hProp _ _) _). - apply isapropdirprod. + apply isaprop_isfilter_imply. + apply isaprop_isfilter_finite_intersection. - simpl. apply filterbase_genetated. + intros A B. apply (BaseOfPreFilter_and base). + apply (BaseOfPreFilter_notempty base). Qed.
Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.