Subband of Induced Operation is Set of Subbands

Theorem
Let $\left({S, \circ}\right)$ be a band.

Let $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$ be the algebraic structure consisting of the power set of $S$ and the operation induced on $\mathcal P \left({S}\right)$ by $\circ$.

Let $T \subseteq \mathcal P \left({S}\right)$.

Then $\left({T, \circ_\mathcal P}\right)$ is a subband of $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$ only if every element of $T$ is a subband of $\left({S, \circ}\right)$.