What is wrong with this proof $M\times \mathbb{R}^n$ diffeomorphic to $TM$

Question

$TM$ is the tangent bundle. Since it is defined as the disjoint union, it is the same set as $M\times \mathbb{R}^n$. Just construct the function $$\begin{align*} f: M\times \mathbb R^n\, &\longrightarrow\, TM\\ (x,u)\,&\longrightarrow (x,u). \end{align*}$$ Since the coordinate chart on $TM$ is defined as $(U,\varphi)$ where $U=\pi^{-1}(V)$ for $V$ open in $M$ and $$ \varphi\left(v^i\frac{\partial}{\partial x_i}\right)= (x,u), $$ it looks like the function constructed above is identity with respect to this coordinate chart. I feel like the tangent bundle is trying to capture the relation between the point and the corresponding tangent space, but since the tangent bundle is defined as a disjoint union, it seems like it is the same as cartesian product.