Powering-invariance is strongly intersection-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., powering-invariant subgroup) satisfying a subgroup metaproperty (i.e., strongly intersection-closed subgroup property)
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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