Imagine I have a set $\mathbb{P}$ with subsets $\mathbb{P_1}, \mathbb{P_2}$, and a multiset $D$.
I need to write that one of the subsets bijects the multiset and the other one does not belong to the multiset not taken from the first one. So, an example:
$$\mathbb{P_1} = \{1, 2\} \longmapsto \left \{-2, 5, -3, \frac{8}{3}, -2 \right \}$$
Suppose $1$ takes $-2$ and $2$ takes $5$, then
$$\mathbb{P_2} = 3, 4, 5 \not \in \left \{-3, \frac{8}{3}, -2 \right \}$$
At first glance, I think this can be written as
$$\begin{cases} \mathbb{P_1} \longmapsto D \\ \mathbb{P_2} ? \left (D - \mathbb{P_1}\right)\end{cases} $$
How one may write it?
Note that $\not \in$ and $\not \longmapsto$ do not work here. $5 \in D$ but $\not \in \mathbb{P_2}$, and saying it's not bijective can lead to various of wrong types of function.