Fiber Product is an embedded submanifold

Question

Let $\pi:E\to B$ a smooth submersion and $\phi:F\to B$ a smooth map. Defining the fiber product of $E$ and $F$ with respect to $B$ as the set:

$$E\times_B F:=\{(e,f)\in E\times F\mid \pi(e)=\phi(f)\}$$

Prove that $E\times_B F$ is an embedded submanifold of $E\times F$.

I'm trying to find charts $\psi:U\cap (E\times_B F)\to\mathbb{R}^k$ where $U\subset E\times F$ is open, but I'm stuck since I don't even know where to look for the right $k$, which makes me think there is probably a simpler solution.

Any suggestions?

Answer 1

A submersion is locally a projection, i.e. around any $e\in E$ there is a neighbourhood diffeo to some $E'\times B'$, such that $\pi$ is just a projection map to $B'\subseteq B$ in these co-ordinates. This reduces to the case where $\pi$ is a projection, which is easy.