See wikipedia. In some sense is a version of the Implicit Function Theorem.
Theorem
Lef $f:M\to N$ a smooth map between manifolds, and let $y\in N$ be a regular value of $f$. Then $f^{-1}(y)$ is a embedded manifold of $M$.
$\blacksquare$
It is a corollary of the Constant rank level set theorem (see @lee2013smooth page 113):
Theorem
Let $M$ and $N$ be smooth manifolds and $F$ a smooth map between them, with constant rank $k$. Each level set of $F$ is a closed embedded manifold of codimension $k$ in $M$.
$\blacksquare$
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Author of the notes: Antonio J. Pan-Collantes
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