# What is the image of a choice function for a family of sets called?

If for any family of sets $$\mathcal{F}$$ we call $$S$$ a choice set of $$\mathcal{F}$$ iff there is a map $$f:\mathcal{F}\to\cup_{S\in\mathcal{F}}S$$ such that $$\forall X\in\mathcal{F}(f(X)\in X)$$ and $$f[\mathcal{F}]=S$$ we see the choice sets of any family $$\mathcal{F}$$ are just the images of choice functions of $$\mathcal{F}$$ - now with that said do these "choice sets" of $$\mathcal{F}$$ already go by another name?

Update: The choice sets of a family of sets $$\mathcal{F}$$ are always the feasible sets of an interval greedoid, more specifically when $$M_{{\small \mathcal{F}}}=\{X\in\mathcal{F}:\forall S\in\mathcal{F}(S\not\subset X)\}$$ is the family of all minimal sets in $$\mathcal{F}$$ where $$T_{{\small \mathcal{F}}}$$ is the family of all bases for the transversal matroid of $$\mathcal{F}$$, so if $$\small [A,B]=\{S:A\subseteq S\subseteq B\}$$, then $$\small C=\{\phi[\mathcal{F}]:\phi\text{ is a choice function of }\mathcal{F}\}=\cup_{(X,Y)\in b(M_{{\small \mathcal{F}}})\times T_{{\small \mathcal{F}}}}[X,Y]$$ is exactly the choice sets of $$\small\mathcal{F}$$ i.e. choice sets of $$\mathcal{F}$$ will form the interval greedoid whose family of minimal feasible sets is $$b(M_{{\small \mathcal{F}}})$$ (the blocker of $$M_{{\small \mathcal{F}}}$$) and whose maximal feasible sets are bases of the transversal matroid on $$\mathcal{F}$$. Don't see the point in writing this as an answer myself since Asaf devoted time so I'm accepting it.

Let me answer from the perspective of a set theorist. Since every set can be the image of a choice function (e.g. if $$A$$ is a set, it is the image of the only choice function from $$\{\{a\}\mid a\in A\}$$), so I am not aware of any particular terminology.