I am having trouble following the identifications in Lee's proof of the following:
Let $V$ be a smooth vector field on a smooth manifold $M$, and let $p\in M$ be a regular point for $V$ . There exist coordinates $(x_i)$ on some neighborhood of $p$ in which $V$ has the coordinate expression $\partial/\partial x^1$.
I have seen the proof using existence/uniqueness of ODE's, and would like to understand it using flows.
Take $p\in M$ a regular point and choose a chart $(y,U)$ about $p$.Thus, $y(p)=0$ and if $q\in U,$ then $y(q)=(y^1(q),\cdots, y^n(q))\in \mathbb R^n$. Wlog $V_p=\left (\partial/\partial y_1\right )_p$.
Let $\theta : (-\epsilon,\epsilon) \times U_0\subseteq \mathcal D(V )\to U\subseteq M$ be the flow of $V.$ So far so good.
Now define $S\subseteq \mathbb R^{n-1}$ by $S=\left \{ (x^2,\cdots, x^n):(0,x^2,\cdots, x^n)\in U_0 \right \}$,
so I assume that this really means that
$S=\left \{ (x^2,\cdots, x^n):y^{-1}((0,x^2,\cdots, x^n))\in U_0 \right \}$.
Now set $ \psi(t, x^2,\cdots, x^n) = \theta (t, (0, x^2,\cdots ,x^n)),$ which really means then that
$ \psi(t, x^2,\cdots, x^n) = \theta (t, y^{-1}((0, x^2,\cdots ,x^n))).$
It is clear that $\psi_*(\partial/\partial t)|_{(t_0,x_0)}=V_{\psi(t_0,x_0)}$ so to finish it suffices to show that $\psi_*$ is an isomorphsim: $T_{(0,0)}((-\epsilon,\epsilon) \times U_0)\to T_pM,$ because in this case, $\psi$ will be a diffoemorphism and so $\psi^{-1}$ will serve as the desired chart.
We have $ \psi(0, x^2,\cdots, x^n) = \theta (0, y^{-1}((0, x^2,\cdots ,x^n)))= y^{-1}((0, x^2,\cdots ,x^n)) $ because $\theta$ is a flow, so
$\psi_*(\partial/\partial x^i)_{(0,0)}f=\frac{\partial}{\partial x^i}(f\circ \psi)(0,0)=\frac{\partial}{\partial x^i}(f\circ y^{-1})((0, x^2=0,\cdots ,x^n=0))=\left (\frac{\partial f}{\partial y^i}\right )_p,$ the last equality is true by definition of $\partial/\partial y^i$. So $\psi_*$ carries basis elements to basis elements, and it follows that $\psi_*$ is an isomorphism.
Is this correct? Where is the fact that $V_p=\left (\partial/\partial y_1\right )_p$ used?