Let $S$ be some set and consider $X \subseteq S$ of size $|X|=x$ u.a.r. (among all the subsets having this size).
Now, use some properties of this set $X$ to find some subset $Y\subseteq X$ of some (variable) size $y$.
When we know want to argue about $Y$, can we argue that it is a random set, when we do not reuse the information for splitting $X$ into $X\setminus Y$ and $Y$?
MORE BACKGROUND:
Consider a "random" graph $G=(V,E)$ and choose $k$ many subsets $X_1, \dotsc, X_k$ u.a.r. with $|X_1|=|X_2|= \dotsc =|X_k|=x$, all disjoint. Now consider $X_i$. Depending on the edges from $X_1, \dotsc, X_{i-1}$ into $X_i$, we split $X_i$ into two parts $X_i \setminus Y_i$ and $Y_i$. When we now argue about edges going from $Y_i$ to $X_{i+1}, \dotsc, X_{k}$, can we say that it is the same as if we have chosen $Y_i$ u.a.r., when all edges are independent?