Section PartitionImage.
Variables (T : finType) (P : {set {set T}}) (D : {set T}).
Variables (T' : finType) (f : T -> T') (inj_f : injective f).
Let fP := [set f @: (B : {set T}) | B in P].

Lemma imset_partition : partition fP (f @: D) = partition P D.