Context {T : Type}.

(* Specifications: *)
Lemma iota_filterlim_locally {F} {FF : ProperFilter' F} (f : T -> V) l :
filterlim f F (locally l) ->
iota (fun x => filterlim f F (locally x)) = l.
Lemma iota_filterlimi_locally {F} {FF : ProperFilter' F} (f : T -> V -> Prop) l :
F (fun x => forall y1 y2, f x y1 -> f x y2 -> y1 = y2) ->
filterlimi f F (locally l) ->
iota (fun x => filterlimi f F (locally x)) = l.