Let $f:X\to Y$ be a birational morphism of smooth projective variety. We assume that $f(V)\simeq U$ isomorphism induced by $f$, where $V\subset X$ and $U\subset Y$ are two Zariski open sets. Let $x\in V$, $C$ be a curve passing through $x$ in $X$ and $L$ be a line bundle over $Y$. Then is the following true?

$$f^{*}L\cdot C=L\cdot\overline{f(C\cap V)}$$

where $f^{*}L$ denotes the pullback of the line bundle $L$ and $\overline{f(C\cap V)}$ is the Zariski closure of $f(C\cap V)$ in $Y$.

Any suggestions/comments are welcome!