Suppose you're working over an algebraically closed field $F$, and let $X$ and $X'$ be quasi-affine varieties, with $Y\subseteq X$ and $Y\subseteq X'$ closed, irreducible subvarieties.
I read the following criterion that the local rings $\mathcal{O}_{Y,X}\cong\mathcal{O}_{Y',X'}$ are isomorphic (as $F$-algebras) iff there exist nbhds $U$ of $Y$, and $V$ of $Y'$ and an isomorphism $\varphi\colon U\to V$ such that $$ \varphi(U\cap Y)=V\cap Y'. $$
(Is there a reason why it's expressed in terms of $U\cap Y$ and $V\cap Y'$, wouldn't these just be $Y$ and $Y'$ themselves, respectively?) Anyway, is there a proof of this criterion out there? The closest I could find was Exercise 1.4.7 on page 30 in Hartshorne, but that only looks at single points inside varieties. Thanks.