Indexed union

Set

 context $f:I\to X$ definiendum $\bigcup_{i\in I,\ f} X_i \equiv \bigcup \mathrm{im}(f)$

Discussion

For $f=\mathrm{id}$, the set is indexing itself and the indexed union is just the arbitrary union $\bigcup_{i\in I} X_i = \bigcup X$.