Section EqTheory.
Variables (aT : finType) (rT : eqType).
Notation fT := {ffun aT -> rT}.
Implicit Types (y : rT) (D : {pred aT}) (R : {pred rT}) (f : fT).
Definition pfamily_mem y mD (mF : aT -> mem_pred rT) :=
family (fun i : aT => if in_mem i mD then pred_of_simpl (mF i) else pred1 y).

Lemma pfamilyP (pT : predType rT) y D (F : aT -> pT) f :
reflect (y.-support f \subset D /\ {in D, forall x, f x \in F x})
(f \in pfamily_mem y (mem D) (fmem F)).