$2$ questions in proof of Proposition $4.6$ of algebraic geometry by Daniel Perrin(page $46$)

I have 2 question in the proof of Proposition $4.6$ of algebraic geometry by Daniel Perrin.( Page $46$).

Statement of Proposition $4.6$: Let $X$ be an algebraic variety and let $Y$ be a closed set in $X$. We define a sheaf of rings $O_Y$ of $Y$ by setting $O_Y(V)= ${$f:V\to k| \forall x\in V \exists U \subset X$, open with $x\in U$ and $g\in O_X(U) $ such that $g|_{U \cap V}= f|_{U \cap V}$} for any open $V$ in $Y$. If $X$ is an algebraic variety ( resp. an affine algebraic variety), then the same is true of $Y$ with the sheaf $O_Y$ and the inclusion of $Y$ in $X$ is a morphism.

Proof. It will be enough to prove the result for affine $X$ since general case immediately follows. We assume that $X$ is affine: it will then be enough to show that the sheaf $O_Y$ is equal to the sheaf $R_Y$ of regular functions on $Y$.Consider $f\in \Gamma(X)$ and it's image in $\Gamma(y), \overline f$. By Lemma 2.2( Statement : Let $X$ be a topological space equipped with a basis of open sets $U$, let $F$ be a sheaf and $G$ be a presheaf on $X$. We assume that $F(U)=G(U)$ for every $U \in U$. Then $F=G^+$.) it will be enough to show that $R(D(\overline {f}))=O_{0,Y}(D(\overline{f})).$ Question: Why it is enough to show this?

Since we know that $D(\overline {f})=D(f) \cap Y$ and the restriction homomorphism $\Gamma(X)_f \to \Gamma(Y)_{\overline{f}}$ is surjective ( Question:How it is surjective?) ....

Rest of the proof is clear to me.

Can you please help me with these two questions?