Let $R$ be a discrete valuation ring with uniformizing parameter $t$, i.e. $\mathfrak{m}_R=(t)$. Let $\widehat{R}$ denote the completion of $R$ with respect to $(t)$. Let $Y$ be a flat locally Noetherian $R$-scheme and denote $\widehat{Y}:= Y \times_{\text{Spec } R} \text{Spec } \widehat{R}$. An ideal sheaf $\mathcal{I} \subset \mathcal{O}_{\widehat{Y}}$ gives rise to a closed subscheme $V(\mathcal{I}) \subset \mathcal{I}_{\widehat{Y}}$. Assume that $V(\mathcal{I})$ is contained in $V(t)$.
Let $\pi : \widehat{Y} \to Y$ be the canonical morphism. As $Y$ is flat over $R$, the canonical homomorphism $\mathcal{O}_Y \to \pi_*\mathcal{O}_{\widehat{Y}} = \mathcal{O}_{Y} \otimes_R \widehat{R}$ stays injective. Thus the definition $\mathcal{I}_0:= \pi_* \mathcal{I} \cap \mathcal{O}_Y$ make sense.
Q: Why does the assumption $V(\mathcal{I}) \subset V(t)$ imply that $\mathcal{I}= \pi^*\mathcal{I}_0$?