Does there exist a model $V$ of $ZFC$ with the following property?
Suppose that $X$ is a set and $\mathcal{A}\subseteq P(X)$ is a collection of subsets such that $X\in\mathcal{A}$ and where $\mathcal{A}$ is a join-semilattice with respect to $\subseteq$ and where $\mathcal{A}$ has a smallest subset with respect to $\subseteq$. Then there is some $Y$ and bijection $f:X\rightarrow Y$ such that $A\in\mathcal{A}$ if and only if there is some ground model $W\subseteq V$ where $Y\cap W=f[A]$.
If not, then what extra conditions on $\mathcal{A}$ should one impose?