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Let $V_K,V_{K_v}$ be the Zariski closures of images of $\Phi$ and $\Phi|_{X(K)}$, respectively. Since $V_{K_v}$ has higher dimension than $V_K$, it is a proper subvariety, which means there must be some nonzero regular function $f$ on $V_{K_v}$ which vanishes on $V_K$. Since $X(K_v)$ is Zariski dense in $V_{K_v}$ (by definition), $f$ must be nonzero on some element of $\Phi(X(K_v))$. That is to say, $f\circ\Phi$ is nonzero on $X(K_v)$, but it will vanish on $X(K)$. Hence $f\circ\Phi$ is the desired function.