Characterizing the Slice Chart of a Subbundle

A rank-$k$ subbundle $F$ of a rank-$n$ smooth vector bundle $E$ is a vector bundle which is smoothly embedded in $E$, whose intersection with a given fiber of $E$ is a subspace of that fiber.

Can we show that, since the subbundle is a smooth embedded submanifold, it has an atlas of slice charts $(\pi^{-1}(U_\alpha),\Phi_\alpha)$ in the ambient bundle satisfying $$\Phi_\alpha(V)=(\widehat{\pi(V)},V^1,\cdots,V^n)$$ Where $V^k=\cdots=V^n=0$ if and only if $V\in F$?