Let $M$ and $N$ be smooth manifolds with Poisson-structures $\{ \cdot , \cdot\}|_M$ and $\{\cdot , \cdot \}|_N$ We call $\phi: M \to N$. a Poisson-map, if the pullback of $\phi$ is compatible with the Poisson-structures, so $ \forall f,g \in C^\infty(N)$ and $ \forall m \in M $ we have $ \{\phi^*f, \phi^*g\}|_M (m) = \phi^*\{f,g\}|_N (m)$.
We call a submanifold $Z \subset N$ coisotropic, if for all $f,g \in C^\infty (N)$ with $f|_Z =g|_Z = 0$ there Poisson-bracket also vanishes on $Z$ $(\{f,g\}|_N(z)=0 \ \ \forall z \in Z)$.
We say the map $\phi : M \to N $ intersects $Z \subset N $ cleanly, if $\phi^{-1}(Z)$ is a submanifold of $M$ and $T_m \phi^{-1}(Z)= (d_m\phi)^{-1}(T_{\phi(m)}Z)$.
My question is now, how to see that, if my Poisson-map $\phi: M \to N$ intersects the coisotropic submanifold $Z \subset N$ cleanly, the preimage $\phi^{-1}(Z)$ is also coisotropic.
