I am reading Silverman's the Arithmetic of Elliptic curves. On page 9, he gave the following definition of projective closure of $V$:
Definition. Let $V\subset \mathbb{A}^{n}$ be an affine algebraic set with ideal $I(V)$, and consider $V$ as a subset of $\mathbb{P}^{n}$ via $$V\subset \mathbb{A}^{n}\xrightarrow{\phi_{i}} \mathbb{P}^{n}.$$ The projective closure of $V$, denoted $\overline{V}$, is the projective algebraic set whose homogeneous ideal $I(\overline{V})$ is generated by $$\{f^{*}(X):f\in I(V)\}.$$
In order for this definition to make sense, I must know the following result: there is a unique projective algebraic set $W$ such that $I(W)=(f^{*}(X):f\in I(V))$. But I don't know how to prove the result. Or do I misunderstand the definition?