stronger collection

czf.v

@@ -311,3 +311,19 @@ Proof.
   2: { rewrite Hx. apply (proj2_sig Hϕ'). }
   exists x. cbn. reflexivity.
 Qed.
+
+Theorem CZF_collection_stong (a : V) (ϕ : V → V → Prop)
+        `{Proper _ ((≡) ==> (≡) ==> (impl)) ϕ} :
+  (forall (x : V), x ∈ a → { y | ϕ x y }) →
+  exists (b : V), (forall (y : V), y ∈ b → exists x, x ∈ a ∧ ϕ x y ).
+Proof.
+  intros Hϕ.
+  exists (coll a ϕ Hϕ). intros yy Hyy.
+  destruct a as [A f].
+  destruct Hyy as [y Hy]. simpl in *.
+  exists (f y). split.
+  { exists y. reflexivity. }
+  rewrite Hy.
+  set (Hϕ':=Hϕ (f y) (ex_intro (λ z, f y ≡ f z) y (reflexivity _))).
+  apply (proj2_sig Hϕ').
+Qed.