I am reading Silverman's book the arithmetic of elliptic curves. He defines rational map as follows:
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, $$ \phi(P)=[f_{0}(P),\cdots,f_{n}(P)]\in V_{2}. $$
Then he also defines $\phi$ being regular at $P\in V_{1}$:
Definition. A rational map $\phi=[f_{0},\cdots,f_{n}]:V_{1}\to V_{2}$ is regular at $P\in V_{1}$ if there is a function $g\in \overline{K}(V_{1})$ such that (i) each $gf_{i}$ is regular at $P$; (ii) there is some $i$ for which $(gf_{i})(P)\neq 0$.
After this definition, he said: if such a $g$ exists, then we set $\phi(P)=[(gf_{0})(P),\cdots,(gf_{n})(P)]$.
My main question is: (1) if a rational map $\phi=[f_{0},\cdots,f_{n}]:V_{1}\to V_{2}$ is regular at $P$, do we have $[(gf_{0})(P),\cdots,(gf_{n})(P)]\in V_{2}$? Why?
This is probably true based on the definition of isomorphism of varieties. But I want to know why. I also want to make sure I understand the definitions correctly.
(2) If I want to check $[f_{0},\cdots,f_{n}]$ is a rational map, I just need to check $f_{0},\cdots,f_{n}\in \overline{K}(V_{1})$, and for every $P\in V_{1}$ at which $f_{0},\cdots,f_{n}$ are all defined and $f_{i}(P)\neq 0$ for some $i$, we have $[f_{0}(P),\cdots,f_{n}(P)]\in V_{2}$. Right?
(3) If $\phi$ is a rational map and I want to check $\phi$ is regular at $P\in V_{1}$, I just need to find some $g\in\overline{K}(V_{1})$ such that (i) and (ii) are true. Right? Do I also need to check that $[(gf_{0})(P),\cdots,(gf_{n})(P)]\in V_{2}$?
I ask this because in his proof of proposition 2.1, he only checked (i) and (ii) but did not check $[(gf_{0})(P),\cdots,(gf_{n})(P)]\in V_{2}$. I'm not sure if he skipped it or we don't need to check.
Thank you for your help!