Section Transversals.
Variables (X : {set T}) (P : {set {set T}}) (D : {set T}).
Hypothesis trPX : is_transversal X P D.
Let tiP : trivIset P. Proof. by case/andP: trPX => /and3P[]. Qed.
Let sXP : {subset X <= cover P}.
End Transversals.
End Partitions.
Arguments trivIsetP {T P}.
Arguments big_trivIset_cond [T R idx op] P [K E].
Arguments set_partition_big_cond [T R idx op] P [D K E].
Arguments big_trivIset [T R idx op] P [E].
Arguments set_partition_big [T R idx op] P [D E].
Prenex Implicits cover trivIset partition pblock.

Lemma indexed_partition (I T : finType) (J : {pred I}) (B : I -> {set T}) :
let P := [set B i | i in J] in
{in J &, forall i j : I, j != i -> [disjoint B i & B j]} ->
(forall i : I, J i -> B i != set0) -> partition P (cover P) /\ {in J &, injective B}.