This may be a stupid question. Suppose some open set $U$ on a manifold is described by local coordinates $(x,y)$. Suppose furthermore there is another coordinate chart $(\tilde{x}, \tilde{y})$ such that
$$\tilde{x} \equiv \tilde{x}(x,y) \quad , \qquad \tilde{y} \equiv \tilde{y}(y)$$
I.e. the coordinate $\tilde{y}$ is $x$-independent. Does this fact alone imply that $y \equiv y(\tilde{y})$ (case 1) or is $y \equiv y(\tilde{x}, \tilde{y})$ possible (case 2)?
My guess is that case 2 is true. Since it is not the map $y \mapsto \tilde{y}(y)$ that must be unique but rather the map $(x,y) \mapsto (\tilde{x},\tilde{y})$. Although $\tilde{y}$ is $x$-independent, the map $y \mapsto \tilde{y}$ may not be injective, meaning that there does not necessarily exist an inverse map $\tilde{y} \mapsto y$, at least not all of $U$.
Is this reasoning correct or is it flawed?