Question regarding syntax of $\bigcap$ definition I know what this definition is supposed to do. However just looking at the syntax, i cannot explain it to myself.
$$
\bigcap M := \{x \mid \forall y \in M : x \in y\}
$$
Could somebody please explain me how to read this definition (not what it does, but rather how to read it close to the definition's syntax so that it is easier to understand)?

$$
\begin{array}{cccccc}
\bigcap M
 & := 
 & \{
 & x 
 & \mid
 & \forall
 &  y  \in M : 
 &  x \in y
 & \}  \\
\text{This “$\bigcap M$” thing}
 & \text{means}
 & \text{the set of}
 & \text{all $x$} 
 & \text{such that}
 & \text{for every}
 & \text{ set $y$ in $M$,}
 & \text{$x$ is in set $y$}
 & \text{.}
\end{array}
$$
