Intersection of subgroups is a subgroup

Proposition (Intersection of subgroups). Let $G$ be a group and let $\{H_i\}_{i\in I}$ be a family of subgroups of $G$. Then the set-theoretic intersection

is a subgroup of $G$.

Context. This implies there is a smallest subgroup of $G$ containing any given subset (the intersection of all subgroups containing it), which underlies the notion of a generated subgroup.

Proof sketch. Each $H_i$ contains the identity, so $H$ is nonempty. If $x,y\in H$, then $x,y\in H_i$ for all $i$, hence $xy^{-1}\in H_i$ for all $i$, so $xy^{-1}\in H$. Apply the one-step subgroup test.