Section in_imfset.
Variable (key : unit) (K : choiceType).
Implicit Types (A B : {fset K}) (a b : K).
Hint Resolve fsubset_refl : core.
Definition fincl A B (AsubB : A `<=` B) (a : A) : B :=
[` (fsubsetP AsubB) _ (valP a)].
Definition fsub B A : {set B} := [set x : B | val x \in A].
Hint Resolve fsub_inj : core.

Lemma fproper_sub_trans B A C : A `<` B -> B `<=` C -> A `<` C.