a condition for smooth vector field

Question

Let $M$ be a Hausdorff manifold. I'm trying to prove that a vector field $Y:M\to TM$ is smooth if and only if the derivation induced by $Y$ for all globablly defined smooth functions is smooth. That is, $Yf:M\to \mathbb{R}$ is smooth for all $f\in C_M^\infty$. In Lee, there is a lemma saying $Y$ is smooth iff for every open set $U\subset M$ and every $f\in C_U^\infty$, the function $Yf:U\to \mathbb{R}$ is smooth. One direction of the proof involves taking coordinate functions of a chart of $U$. I'm not sure how to do this for the global case.