Concerning families of sets regarded as functions

In the course I am taking of axiomatic set theory we've defined a family of sets $F$ indexed by $I$ as any function satisfying $dom(F)=I$, where there are no assumptions about its image. This definition confuses me a little because I've read, in other contexts, that families of sets differ from sets because the first can be proper classes while the later obviously can't, but with this definition $Im(F)$ must be a set since it is a subset of $\bigcup \bigcup F$

Could anyone clarify it? Thanks

What you defined as 'family of sets' is more commonly known as a 'sequence'. Let's write $F_i := F(i)$ for each $i \in I$. Then your family of sets $F$ is precisely the sequence $F = (F_i \mid i \in I)$ and you're right in that, that - following your definition and working in a sufficiently strong theory, e.g. ZFC - this is always a set.
Other authors use the term 'family of sets' differently and to them it means just 'a collection of sets' that does not neccessarily have to be a set itself. An example is the collection of all groups or the collection of the von Neumann hierarchy $\{ V_\alpha \mid \alpha \in \operatorname{On} \}$.