How can we formally define a Pseudo-Random Shuffle function? My question is related to: When we turn Random shuffle to Pseudorandom Shuffle

The idea is to permute the elements in vector $\mathbf{v}$ pseudorandomly, where $|\mathbf{v}|=n$;

I am aware that we can use Fisher-Yates shuffle for pseudorandom shuffle (in fact we replace the random generator with a pseudorandom one). So we can use a key: $k$, when we want to pseudorandomly shuffle the vector. 

Let the pseudorandom shuffle be: $\pi (k,\mathbf{v})$


How to formally define the pseudorandom function? For instance, do we need to say pseudorandom shuffle must be indistinguishable from random shuffle?
Is there any textbook or paper has already defined it?
A PRP (pseudorandom permutation) can be mapped onto the index of each element in vector $\mathbf{v}$ to generate a new index for that element in the shuffled array.

As long as the PRP is secure, the shuffle should be indistinguishable from random. Theoretically, this works if the PRP's input/output is the same size as the number of elements in the array. But in the real world, a PRP construction that can vary it's input/output size for various lengths doesn't exist.

AES-256 could be used as the PRP if there were $2^{256}$ elements in the array. Any stream cipher could be used as long as the size of the vector was a power of $2$. In the real world, there are also memory issues as you have to store values somewhere as you shuffle them. Fisher-Yates solves both these issues by using a PRNG to choose elements to exchange.

That's just my idea of a way to define a pseudorandom shuffle. Using a PRP should make the proof easy. Here's a paper that defines a pseudorandom shuffle in a different way: "The Strength of Weak Randomization: EasilyDeployable, Efficiently Searchable Encryptionwith Minimal Leakage"