Proof that the rank of a differentiable function on a manifold is well-defined On p. 40, Spivak's Differential Geometry vol. 1 makes the following claim:

Now, if $f: M^n \rightarrow N^m$ is $C^\infty$ and $(y, V)$ is a coordinate system around $f(p)$ [and $(x, U)$ is a coordinate system around $p$], the rank of the $m \times n$ matrix
\begin{align}
\left(\frac{\partial(y^i \circ f)}{\partial x^j}(p) \right)
\end{align}
clearly does not depend on the coordinate system $(x, U)$ or $(y, V)$.

However, I don't see why this should be true (aside from some vague intuitions about linear algebra), and I can't even think where to start on a proof. So, what does a proof of this claim look like, or at least, why it is "clearly" true?
 A: The matrix
$$ J_{x,y}f(p) := \left(\frac{\partial(y^i\circ f)}{\partial x^j}(p)\right)_{\begin{array}{l}i=1\ldots m\\[-1ex]j=1\ldots n\end{array}} $$
is the jacobian matrix of $f$ at $p$ with respect to the coordinate systems, $x$ and $y$, we have chosen. It represents the differential $df(p)\;:\;T_p(M)\to T_{f(p)}(N)$ with respect to their canonical bases, $\left(\frac{\partial\,\cdot}{\partial x^j}(p)\right)_{j=1\ldots n}$ of $T_p(M)$, and $\left(\frac{\partial\,\cdot}{\partial y^i}(f(p))\right)_{i=1\ldots m}$ of $T_{f(p)}(N).$
We recall that
$$ J_{x,y}f(p) = \Big(D_j(y^i\circ f\circ x^{-1})(x(p))\Big)_{\begin{array}{l}i=1\ldots m\\[-1ex]j=1\ldots n\end{array}} = J(y\circ f\circ x^{-1})(x(p)), $$
where the $J$ on the right is the jacobian for functions from (restrictions to open subsets of) $\mathbb R^m$ to $\mathbb R^n$, like $y\circ f\circ x^{-1}\;$ is (and where $D_j$ is the usual partial derivative with respect to $j$-th real variable).
If we choose two other coordinate systems, $\tilde x$ around $p$ and $\tilde y$ around $f(p)$, and remembering that the jacobian of a composition is the product of the jacobians, then we have
\begin{align}
J_{\tilde x,\tilde y}f(p) &= J(\tilde y\circ f\circ \tilde x^{-1})(\tilde x(p)) =\\[1ex]
&= J\Big((\tilde y\circ y^{-1})\circ(y\circ f\circ x^{-1})\circ(x\circ\tilde x^{-1})\Big)(\tilde x(p)) =\\[1ex]
&= J(\tilde y\circ y^{-1})(y(f(p)))\cdot J(y\circ f\circ x^{-1})(x(p))\cdot J(x\circ\tilde x^{-1})(\tilde x(p)) = \tag{*}\\[2ex]
&= J(\tilde y\circ y^{-1})(y(f(p)))\cdot J_{x,y}f(p)\cdot J(x\circ\tilde x^{-1})(\tilde x(p))\\
\end{align}
from which the conclusion as $\tilde y\circ y^{-1}$ and $x\circ \tilde x^{-1}$ being diffeomorphisms, their jacobians don't alter the rank of the central term.
Please note: for the sake of brevity in (*) we have omitted the restrictions. For example we should have written $\tilde y\circ y^{-1}\Big|y\big(\text{Domain}(\tilde y)\cap\text{Domain}(y)\big)$
instead of $\tilde y\circ y^{-1},\;$ etc.
