Proposition lam_image_form_eq {Γ : ty H} (x : tm Γ X) (y : tm Γ Y) (z : tm Γ Z) : ⟨ x , y ⟩ ∈ {{lam_image_form}} [⟨ !!, z ⟩ ]tm = φ [ ⟨ ⟨ x , z ⟩ , y ⟩ ]. Proof. refine (!_). etrans. { refine (_ @ maponpaths (λ φ, φ [ ⟨ ⟨ x , y ⟩ , ⟨ !! , z ⟩ ⟩ ]) lam_image_form_eq_help). unfold lam_image_form. cbn. simplify. apply idpath. } cbn. simplify. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Hyperdoctrines.Hyperdoctrine. Require Import UniMath.CategoryTheory.Hyperdoctrines.FirstOrderHyperdoctrine. Require Import UniMath.CategoryTheory.Hyperdoctrines.Tripos. Require Import UniMath.CategoryTheory.Hyperdoctrines.GenericPredicate. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERs. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERMorphisms. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERCategory. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERBinProducts. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.ExponentialPER.