Let $S$ be a set. If there exists a set of the coordinate system $A$ of $S$ which satisfies the below conditions then we say that $S$ is an n-dimensional $C^{\infty}$ manifold.
(1) Each element $\phi$ of $A$ is a one-to-one mapping from $S$ to some open subset of $\mathbb{R^n}$
(2)For all $\phi\in A$, given any one-to-one mapping $\psi$ from $S$ to $\mathbb{R}^n$, the following holds:
$$\psi\in A \iff \psi\circ\phi^{-1} \text{ is a }C^{\infty} \text{ diffeomorphism }$$
Also ,let $S$ have a coordinate $[\xi^i]$ with respect to $\phi:S\to\mathbb{R^n}$ and another coordinate $[\rho^i]$ with respect to coordinate system $\psi:S\to \mathbb{R^n}$ then the transition map is defined as $\psi\circ\phi^{-1}:[\xi^1,...,\xi^n]\to[\rho^1,....,\rho^n]$
The book that I am reading said that from conditions (1) and (2) and by the definition of transition maps we can partially differentiate $\rho^i=\rho^i[\xi^1,....,\xi^n]$ and $\xi^i=\xi^i[\rho^1,....,\rho^n]$, can someone explain how they consider $\rho^i$ as a function of $\xi^i$ and vice-versa?
$\psi \circ \phi^{-1}$ is a function from $\mathbb{R}^n \to \mathbb{R}^n$, and $\rho^i$ is the $i$-th component of this function.