# How can I control the height of my left-right delimited equation part?

I have a simple equation, much like the following (I use the braket package) :

$$\mathcal{Y} = \Set{ x \in \mathcal{G} | \Vert x \Vert > d }$$

My gripe with this equation's rendering is that the | and || are close to eachother and of the exact same height - I'd like the | bar to be bigger to make the symbols more distinct. One way to do this would be to make the whole Set, or left-right environment, a bit taller. How can I do this? Can I insert an invisible character that will make the whole thing bigger? Should I insert a higher | bar manually?

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$$\mathcal{Y} = \Set{ x \in \mathcal{G} | \Vert x \Vert > d }$$
$$\mathcal{Y} = \Set{ x \in \mathcal{G} | \strut\Vert x \Vert > d }$$