Commutator-in-center is intersection-closed

Property-theoretic statement
The property of being a commutator-in-center subgroup is an intersection-closed subgroup property: it is closed under arbitrary nonempty intersections. Note that it is not a strongly intersection-closed subgroup property because it is not closed under the empty intersection, since every group need not satisfy this property within itself.

Statement with symbols
Suppose $$G$$ is a group and $$H_i, i \in I$$ is a nonempty collection of commutator-in-center subgroups of $$G$$. Let $$H = \bigcap_{i \in I} H_i$$. Then, $$H$$ is also a commutator-in-center subgroup of $$G$$.

Related facts

 * Normality is strongly intersection-closed