In the book "Topology from the differential viewpoint" from John Milnor, we have the following theorem,
Let $f: M \rightarrow N$ a smooth map between smooth manifolds of dimensions respectively dim($M$) = $m$ and dim($N$) = $n$. Let $y \in N$ a regular value of $f$.
Then $Z := f^{-1}(\{y\})$ is a submanifold of dimension dim($M$) - dim($N$) and its tangent space $\forall p \in Z$ is given by,
$$ T_p Z = \text{ker}(df_p: T_pM \rightarrow T_yN) $$
I have trouble proving the second part of the theorem with the tangent space. Here's my attempt :
We know that $Z \subset M$ and $f: M \rightarrow N$.
We define the inclusion map: $$ i: Z \hookrightarrow{} N $$
Then, $$ f \circ i = F_{|_{Z}} $$
We wanna show that $T_pZ \subset \ker(df_p: T_xM \rightarrow T_yN)$. So, we know that,
$$ T_pZ := im(d(f_{|_Z})_p: T_pZ \rightarrow T_yN) $$
Then, $$ d(f_{|_Z})_p = d(f \circ i)_p = df_p \circ di_p : T_pZ \hookrightarrow T_pM \rightarrow T_yN $$
Then, $$ im(d(f_{|_Z})_p) = im(df_p \circ di_p) = T_yN \subset T_y\{y\} = im\{0\} $$
So we have, $$ T_pZ \subset im\{0\} $$
Then I'm stuck here. How could I finish the proof ?