Section FsetPartitions.
Variables T I : choiceType.
Implicit Types (x y z : T) (A B D X : {fset T}) (P Q : {fset {fset T}}).
Implicit Types (J : pred I) (F : I -> {fset T}).
Definition fcover P := (\bigcup_(B <- P) B)%fset.
Definition trivIfset P := (\sum_(B <- P) #|` B|)%N == #|` fcover P|.

Lemma fcover_imfset (J : {fset I}) F (P : pred I) :
fcover [fset F i | i in J & P i]%fset = (\bigcup_(i <- J | P i) F i)%fset.