I have shown the following lemma:
If $f:U\subset\mathbb{R}^n\longrightarrow \mathbb{R}^n$ is a $C^1$ map, then if $A\subset U$ is a set of measure zero then $f(A)$ has measure zero.
The idea of the proof I'm following say "Let's find an open set $U$ of $\mathbb{R}^n$ such that $M\subset U$ and $U\cap M$ has measure zero. So there exists $f:U\longrightarrow f(U)$ a $C^1$ diffeomorphism such that $f(U\cap M)=f(U)\cap(\mathbb{R}^{d}\times\{0\}^{n-d})$ so $U\cap M=f^{-1}\Big(f(U)\cap(\mathbb{R}^{d}\times\{0\}^{n-d})\Big)$. So we can conclude with the lemma."
I understand the use of the lemma and the conclusion in general. The thing I don't understand is that the author doesn't explain how it is possible to find such a $U$ and why this $U$ would obey the local chart definition of submanifold.