In (one of) the proofs of the Cantor-Bernstein-Schroeder theorem, one defines recursively a sequence of bijections along with a sequence of disjoint domains and co-domains. The (infinite) union of these functions is again a bijection, and of course its domain and codomain are the unions of the respective recursively defined sequences as well. If we abstract a bit, what happened here is that we had $3$ sequences of sets,
- The sequence of functions $f_n$
- The sequence of domains $D_n$
- The sequence of co-domains, $C_n$
And there was some property, $\alpha(f,X,Y)$ such that for any $n\in \mathbb{N}$,
$$\alpha(f_n,D_n,C_n)$$
Now if we define the union of a tuple as the tuple of the unions, for example (in the case of an ordered pair)
$$\bigcup_{n\in \mathbb{N}} (X_n,Y_n):= \left(\bigcup_{n\in \mathbb{N}} X_n,\bigcup_{n\in \mathbb{N}} Y_n\right)$$
Then we can extract a certain property that was essential for the proof of the above theorem to work, namely that
$$\forall_{n\in \mathbb{N}}[\alpha(f_n,D_n,X_n)]\implies \alpha\left(\bigcup_{n\in \mathbb{N}}(f_n,D_n,C_n)\right) $$
So it was essential that $\alpha$ could be "pulled out of" the union. This is reminiscent of the sequential definition of the continuity of a function, and as I do more mathematics, I find myself needing to prove that this sort of thing is true for a certain property, which motivates me to ask if this is a notion which has been investigated, and if there are useful alternative characterisations of properties which are preserved under this kind of limiting process?