If $S$ is a set, define $$(x \to y) \cdot S := \begin{cases} (S \setminus \{x\}) \cup \{y\} & \text{ if } x \in S \text{ and } y \not \in S; \\ S & \text{ otherwise.} \end{cases}$$
In other words:
If $S$ does not contain $x$, leave it alone.
If $S$ contains $x$ but not $y$, replace $x$ with $y$.
If $S$ contains both $x$ and $y$, though, leave it alone.
Is there a pre-existing notation for this operation anywhere in the literature?
Bonus question: I'd also like a compact notation for $$(x_1 \to y_1) \cdot (x_2 \to y_2) \cdot S,$$ if it exists.