I'm reading from Guillemin and Pollack's Differential Topology, and I don't understand one of the reformulations of the Inverse Function Theorem as well as I'd like to.
The Inverse Function Theorem as stated in G&P is : Suppose $f : X \rightarrow Y$ is a smooth map whose derivative $df_{x}$ at the point $x$ is an isomorphism. Then, the map $f$ is a local diffeomorphism at $x$.
On Page 14 (Chapter 1, section 3) they write "We can suggestively reformulate the Inverse Function Theorem using local coordinates: If $df_{x}$ is an isomorphism then one can choose local coordinates around $x$ and $f(x)$ so that $f$ appears to be the identity $f(x_1, \dots, x_n) = (x_1, \dots, x_n)$"
My intuition behind this reformulation is that if $df_{x}$ is an isomorphism, then $f$ is a local diffeomorphism at that point. The fact that $f$ is a local diffeomorphism means that a neighborhood $V \subseteq X$ around $x$ and a neighborhood $f(V) \subseteq Y$ around $f(x)$ can both be parameterized by the same open set $U \subseteq \mathbb{R^k}$ (assuming $X$ and $Y$ are $k$-dimensional manifolds), and so we are able to impose coordinates on $V$ and $f(V)$ which are in perfect correspondence -- and this makes $f$ behave essentially like the identity map.
I feel like this intuition isn't really rigorous/might be missing something. So, I wanted to ask, is this intuition valid? and are there other ways of thinking about this local coordinates reformulation of the IFT?