Definition:Restriction/Notation

Notation
The use of the symbol $\restriction$ is a recent innovation over the more commonly-encountered $|$.

Thus the notation $\mathcal R |_{X \times Y}$ and $\left({T, \circ|_T}\right)$, etc. are currently more likely to be seen than $\mathcal R {\restriction_{X \times Y}}$ and $\left({T, \circ {\restriction_T}}\right)$.

No doubt as the convention becomes more established, $\restriction$ will develop.

It is strongly arguable that $\restriction$, affectionately known as the harpoon, is preferable to $|$ as the latter is suffering from the potential ambiguity of overuse.

Some authors prefer not to subscript the subset, and render the notation as:
 * $f \mathbin \restriction X = \left\{{\left({x, f \left({x}\right)}\right): x \in X}\right\}$

but this does not conform to house style.