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stronger collection

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Dan Frumin 2019-01-22 13:51:22 +01:00
parent c9f427f737
commit 0578569c09
1 changed files with 16 additions and 0 deletions
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czf.v
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@ -311,3 +311,19 @@ Proof.
2: { rewrite Hx. apply (proj2_sig Hϕ'). } 2: { rewrite Hx. apply (proj2_sig Hϕ'). }
exists x. cbn. reflexivity. exists x. cbn. reflexivity.
Qed. 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.