Let $X$ be a smooth manifold, let $m \in \mathbb{R}$, let $A \in \Psi_h^m(X)$ be a semiclassical pseudodifferential operator, and let $(u_h)_{0<h\le h_0}$ be a semiclassically tempered family of distributions on $X$. Then \begin{align*} \operatorname{WF}_h(Au_h) \subset \operatorname{WF}_h(u_h). \end{align*} Moreover, if $q \in \overline{T}^*X$ and $A$ is elliptic at $q$, then \begin{align*} q \in \operatorname{WF}_h(u_h) \iff q \in \operatorname{WF}_h(Au_h). \end{align*}