I am reading a proof in Pollack on page 71 and in one of the steps in the proof that $N(Y) = \{ (y,v) \in Y \times N_y(Y) \}$ is a manifold, he claims that
$N(U) = N(Y) \cap (U \times \mathbb{R}^M)$ where $U = \tilde{U} \cap Y = \phi^{-1}(0)$. Here $\phi: \tilde{U} \to \mathbb{R}^{codim Y}$ is a submersion and from what I understood, he is defining $Y$ to be the preimage of $0$ of some function and $\phi^{-1}$ just happens to map $0$ to an open set in $Y$.
Now what I don't understand here is the equality $$N(U) = N(Y) \cap (U \times \mathbb{R}^M)$$
I understand the inclusion $\subset $ but not the other.
Both sets are $\{(u,v)|u\in U \text{ and } v\in N_u(Y) \}$.