TITLE: The tangent space of a manifold at a point given as the kernel of the jacobian of a submersion
QUESTION [10 upvotes]: Let $\phi:M\to N$ is a smooth map, $q\in N$ a regular value, and $V=\phi^{-1}(q)$. I want to show that, for each $p\in V$, $T_p(V)= \mathrm{ker}(\phi_*)\subseteq T_p(M)$ (where $\phi_*$ is the differential of $\phi$). 
Since $q$ is regular, I know $\phi_*(T_p(M))=T_q(N)$. I believe I can say $\phi_*(T_p(V))=T_q(\{q\})=0$. But I don't know why $\phi_*^{-1}(0)=T_p(V)$. 
Any input would be greatly appreciated.

REPLY [2 votes]: Hint: Let $j : V \to M$ be the canonical injection. Show that $dj(p) : T_pV \to T_pM$ induces an isomorphism between $T_pV$ and $\mathrm{ker} (d\phi(p))$.