Some books (and actually wikipedia too) define tangent bundle $TM$ of a manifold $M$ as the disjoint union of tangent spaces. Then, they write:
\begin{equation} TM=\bigsqcup_{x\in M} T_xM=\bigcup_{x\in M} \{x\}\times T_xM \tag{1} \end{equation} Where $TM_x$ is the tangent space in the point $x$. So the elements of $TM$ are the couples
\begin{equation} (x,v)\quad x\in M, v\in T_xM \end{equation}
But when I look up the definition of disjoint union, I find something (apparently) different. For example, on wikipedia, the disjoint union of a family of sets $\{A_i, i\in I\}$ (where, as far as understand $I$ index set) is defined as:
\begin{equation} \bigsqcup_{i\in I}A_i=\bigcup_{i\in I}\{(x,i): x\in A_i\} \tag{2} \end{equation}
I can't really find the similarity between $(1)$ and $(2)$. Is the index set replaced by the set of tangent spaces? Also, the subscript of $\bigcup$ is quite different. In $(1)$ we have the points of the manifold while in $(2)$ we have the indexes. Can someone explain to me the relation between these?