# Does a function $f \in W^{n,p}(U)$ which vanishes on $U-\overline{V}$ automatically lives in $W^{n,p}_0 (V)$?

Suppose $$U,V$$ are open sets in $$\mathbb{R}^n$$ and $$\overline{V} \subseteq U$$. Suppose $$f \in W^{n,p} (U)$$ is $$0$$ on $$U-\overline{V}$$. Can we say that the restriction of $$f$$ to $$V$$ (denoted again by $$f$$) is in $$W^{n,p}_0 (V)$$?

I suspect that this is true (at least if $$p \neq \infty$$) since $$W^{n,p}_0$$ intuitively corresponds to functions which are $$0$$ on $$\partial V$$. But I cannot formalize my idea.

Any help will be fully appreciated. Thank you.