Suppose we are given three smooth manifolds $V\subset U$ and $M$, and a smooth function $f\colon U\to M$. Suppose also that $\epsilon\in M$ is a regular value for $f$ (so $W=f^{-1}(\epsilon)$ is a regular submanifold of $U$) and also for $f|_V\colon V\to M$ (so $V\cap W$ is a regular submanifold of $U$).
My question is : In this setting, can we conclude that $V\pitchfork_U W $ ?
The only thing we know by hypothesis is that, for all $x\in W$, we have $f_{\star,x}(T_xU)=T_{f(x)}M$and for all $x\in V\cap W$, we have $f_{\star,x}(T_xV)=T_{f(x)}M$, but I can't find any algebraic argument which would show that for such $x$, we have $T_xV+T_xW=T_xU$. Perhaps just using the codimensions would suffice, but again I have no idea how to proceed. Thank you all for your help!