Why is an "indexed set" indeed a set?

Question

Let $ S = \{S_{i}:i \in I \} $. If $I$ is a set, and for every $ i \in I$, $S_{i}$ is a set. Why does it imply that $S$ must be a set? I have been thinking about this for a while but it's like a loop. I always come to the same point, and nothing is really proven. I do not from where, but I recall that if $f:A \longrightarrow B$ is a surjective function, and $A$ is a set, then $B$ must be a set. Is it helpful here? And if so, what is needed in order to prove it?