Pseudomeromorphic currents on subvarieties

Let $i\text{\hspace{0.17em}⁣}:X\to Y$ be pure-dimensional reduced subvariety of a smooth manifold $Y$. We prove that the direct image of pseudomeromorphic currents on $X$ are pseudomeromorphic on $Y$. We also prove a partial converse: if $i_{\ast }\tau$ is pseudomeromorphic and has the standard extension property, then $\tau$ is pseudomermorphic on $X$.