If a neural network is a function f that maps x to y then is it correct to say that $f:R^x \rightarrow R^y$?

Question

I can say that a simple neural network can be formally defined as $ f: R^D \to R^L$ where $D$ is the size of the input vector $x$ and $L$ the size of the output vector $y$. So I can say that $y = f(x)$.

Now I wonder whether it would also be correct to state the following:

$f: R^x \to R^y$

Would that be formally also correct given the information above? I want to express that this function $f$ maps an input $x \in R^{10}$ to an output $y \in R^{3}$. Is it understandable and formally correct that I want to express with $R^{x}$ that the input is of the dimension like $x$?