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I'm trying to prove the following result: let $f\colon M\to N$ be a surjective submersion between two manifolds, and $E\subseteq N$ an embedded submanifold. Then $S=f^{-1}(E)$ is also an embedded submanifold of $M$ and $f\colon S\to E$ is a submersion.

So far I've been able to prove that $S$ is an embedded submanifold via local coordinates, but I'd be grateful if you could check my reasoning for proving that $f\colon S\to E$ is a submersion.

Let $p\in S$ and $q=f(p)$. Since $f$ is a submersion, there exists a local smooth section $\sigma\colon V\subseteq N\to M$ of $f$ such that $p=\sigma(q)$. Therefore, the restriction $\sigma\colon E\cap V\to M$ is smooth, and $\sigma(E\cap V)\subseteq S$, since for every $z\in E\cap V$ we have $f(\sigma(z))=z\in E$, so $\sigma(z)\in S$. Because of this, and the fact that $S$ is embedded, $\sigma\colon E\cap V\to S$ is a local section of $f\colon S\to E$ such that $\sigma(q)=p$, which implies that $f$ is a submersion at $p$.

Is my reasoning correct? Thank you in advance!

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    $\begingroup$ It falls apart at $q=f(p)$. We have $p \in E \subset N$ which is the range of the function $f$, not the domain, so the evaluation $f(p)$ makes no sense. $\endgroup$ – Lee Mosher Aug 6 at 19:44
  • $\begingroup$ Oh, my mistake. I meant $p\in S$. I'll change it now, thank you for noticing it. $\endgroup$ – Darth Lubinus Aug 6 at 20:22

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