I am reading Silverman's the arithmetic of Elliptic curves. On pages 11 and 12, he gives two definitions of rational maps. I am trying to show these two definitions are equivalent.
The first definition:
Definition. Let $V_{1}$ and $V_{2} \subset \mathbb{P}^{n}$ be projective varieties. A rational map from $V_{1}$ to $V_{2}$ is a map of the form $$ \phi:V_{1}\to V_{2},\quad \phi=[f_{0},\cdots,f_{n}], $$ where the functions $f_{0},\cdots,f_{n}\in \overline{K}(V_{1})$ have the property that for every point $P\in V_{1}$ at which $f_{0},\cdots,f_{n}$ are all defined, $$ \phi(P)=[f_{0}(P),\cdots,f_{n}(P)]\in V_{2}. $$
I think the first definition also works for $V_{1}\subset \mathbb{P}^{m}$, $V_{2}\subset\mathbb{P}^{n}$.
In Silverman, $K$ is a perfect field; $\overline{K}$ is a fixed algebraic closure of $K$; $V\subset \mathbb{P}^{n}$ is a projective variety if $I(V)$ is a prime ideal in $\overline{K}[X_{0},\cdots,X_{n}]$; choose $\mathbb{A}^{n}\subset \mathbb{P}^{n}$ such that $V\cap \mathbb{A}^{n}\neq \emptyset$, then $\overline{K}(V)$ is defined to be $\overline{K}(V\cap \mathbb{A}^{n})$. $\overline{K}(V)$ can also be described as $f(X)/g(X)$ where $f,g\in \overline{K}[X_{0},\cdots,X_{n}]$ are homogeneous of the same degree and $g\notin I(V)$.
The second definition:
Definition. Let $V_{1}\subset \mathbb{P}^{m}$ and $V_{2}\subset \mathbb{P}^{n}$ be projective varieties. A rational map $\phi:V_{1}\to V_{2}$ is a map of the form $$ \phi=[\phi_{0}(X),\cdots,\phi_{n}(X)], $$ where (i) the $\phi_{i}(X)\in\overline{K}[X]=\overline{K}[X_{0},\cdots,X_{m}]$ are homogeneous polynomials, not all in $I(V_{1})$, having the same degree; (ii) for every $f\in I(V_{2})$, $$ f(\phi_{0}(X),\cdots,\phi_{n}(X))\in I(V_{1}). $$
I think I did one direction: if a rational map satisfies the first definition, then it also satisfies the second definition. But I cannot do the other direction: if a rational map satisfies the second definition, then it satisfies the first.
What I have tried so far: let $\phi=[\phi_{0},\cdots,\phi_{n}]$ be a rational map satisfying the second definition. (My plan is to define a $\phi'$, and then show that $\phi'$ satisfies the first definition, and that $\phi=\phi'$.) I tried: we have that $\phi_{i}\notin I(V_{1})$ for some $i$. Let $f_{l}=\phi_{l}/\phi_{i}$ for all $l$, and let $\phi'=[f_{0},\cdots,f_{n}]$. Then I want to check $\phi'$ satisfies the first definition. We have $f_{0},\cdots,f_{n}\in \overline{K}(V_{1})$. Let $P\in V_{1}$ be at which $f_{0},\cdots,f_{n}$ are all defined, then I want to show $[f_{0}(P),\cdots,f_{n}(P)]\in V_{2}$. If $\phi_{i}(P)\neq 0$, then I can prove it by using (ii) from the second definition. But if $\phi_{i}(P)=0$, I don't know how to prove it.
Can someone help me? Thank you.