I'm reading Kanamori's book on large cardinals, and on page 6 he defines truth of $\Delta_0$ formula for transitive proper class M as follows, $\vDash^0_M$ $\phi[x_1,...,x_k]$ iff $\phi(v_1,...,v_k)$ is $\Sigma_0$ and $\exists$ $y\in M$ ($y$ is transitive & $x_1,...,x_k$ $\in$ $y$ & $ \langle y, \in\rangle$ $\vDash \phi[x_1,...,x_k]$ ) My questions are
(1) Why do we require $y \in M$? Because of the absoluteness of $\Delta_0$ formula, isn't the fact that there is some such $y$ sufficient?
(2) Isn't the requirement that $y \in M$ too strong? In general, given that $x_1,...,x_k$ in $M$ and that $\phi^M[x_1,...,x_k]$, how do we infer that there is some $y \in M$ so that $x_1,...,x_k$ are in $y$?
I think I may be missing something simple. Thank you for the help!