**Edited**: Phil Tosteson suggested Thom's first isotopy lemma, but it does not seem to be in the direction that I'm trying to generalize. Let me reformulate my question again.


Let $N\subset M$ be a pair of the smooth manifolds and $\pi:M\to B$ a proper submersion to a smooth base. Assume moreover that the restriction $\pi\_{|N}:N\to B$ is also proper and submersive, then both $\pi$ and $\pi\_{|N}$ are locally trivial by applying Ehresmann's theorem separately, but I'm wondering if they can be *simultaneously* locally trivialized.


More precisely, I'm looking for a proof of the following argument:



> 
> **Claim**: For each $b\in B$, there is a neighborhood $U$ of $b$, and a fiber preserving diffeomorphism 
>  $$\psi:\pi^{-1}(U)\xrightarrow{\cong} M\_b\times U,$$
>  satisfying $$\psi(N\cap \pi^{-1}(U))=N\_b\times U$$
>  where $M\_b=\pi^{-1}(b)$ and $N\_b=\pi\_{|N}^{-1}(b)$.
> 
> 
> 


The example that I have in mind is a smooth family of pairs (cubic surface, cubic curve). Explicitly take a general pencil $\mathbb P^1$ of hyperplane sections of a smooth cubic threefold $X$ over $\mathbb C$ with base locus a cubic curve $E$ (as the intersection of any two hyperplane section in the pencil). Project the incidence variety $I=\{(y,t)|y\in X\cap H\_t\}\subset X\times \mathbb P^1$ to the second factor produces a family of cubic surfaces. Now $M\to B$ will be restriction of $I\to \mathbb P^1$ to the locus where cubic surfaces are smooth and $N:=E\times B$.


(For a smooth family of compact complex manifolds, one can even ask for transversely holomorphic trivialization, see page 2 in [this notes](https://www.math.stonybrook.edu/~cschnell/pdf/notes/kodaira.pdf), I would also like to know if it works for a pair.)



A relative version of Ehresmann's theorem
The answer is positive. There are several proofs of Eheresmann's genuine lemma; I think that each of them can be straightforwardly generalized and gives your relative version. But you can also, alternatively, deduce the relative version from the absolute one together with a classical theorem of Cerf, as follows. After restricting $B$ to a small compact ball centered at a point $b$, by a first application of the genuine Ehresmann lemma to $(M,\pi)$, you can assume without loss of generality that $M=M\_b\times B$ and that $\pi$ is the second projection. Then, after restricting $B$ to a smaller compact ball centered at $b$, by a second application of the genuine Ehresmann lemma to $(N,\pi\vert N)$, there is a local trivialization of $\pi\vert N$ over $B$, in other words a diffeomorphism 
$$\psi\_N:N\to N\_b\times B$$
such that $\pi\circ\psi\_N=\pi\vert N$. You can also see $\psi\_N$ as a parametric family of smooth embeddings $$f\_y:N\_b\hookrightarrow M\_b$$ parametrized by $y\in B$, namely $$(f\_y(x),y)=\psi\_N^{-1}(x,y)$$
It is classical (Cerf) that a parametric family of embeddings of $N\_b$ in $M\_b$ extends to a parametric family of self-diffeomorphisms (isotopies) of the ambiant manifold $M\_b$: there is a smooth family $(F\_y)$ of self-diffeomorphisms of $M\_b$ such that $f\_y=F\_y\vert N\_b$; and $F\_b$ is the identity. Finally, the trivialization $\psi$ that you are looking for is just the inverse of $$(x,y)\mapsto(F\_y(x),y)$$
($x\in M\_b$, $y\in B$)


