Proof of Lemma 6.1, II in Hartshorne

Question

In Hartshorne's proof for Lemma 6.1, II, he said "it would be sufficient to show that there only finitely many prime divisors $Y$ of $U$ for which $v_Y(f) \neq 0$". How can we conclude that $v_Y(f) \neq 0$ for $Y$ is a prime divisor in $X$ if and only if $v_{Y'}(f) \neq 0$, with $Y' = Y \cap U$ ?