Powering-invariance is strongly intersection-closed

Statement
Suppose $$G$$ is a group, $$I$$ is an indexing set, and $$H_i, i \in I$$ is a collection of powering-invariant subgroups of $$G$$. Then, the intersection of subgroups $$\bigcap_{i \in I} H_i$$ is also a powering-invariant subgroup of $$G$$.

Related facts

 * Powering-invariance is transitive
 * Powering-invariance does not satisfy intermediate subgroup condition
 * Powering-invariance is not finite-join-closed