notation : $f_{X|Y=y}(x)$ instead of $f_{X|Y}(x|y)$?

Question

I just have a notation question. The density $f_{X|Y}(\cdot, y)$ is defined by $$\mathbb{P}(X\in A|Y=y)=:\int_A f_{X|Y}(x|y)dx$$ would the notation $f_{X|Y=y}(x)$ instead of $f_{X|Y}(x|y)$ also be reasonable? I only see the first notation used.