Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space.
Let $X \in \mathsf{Random}(P)$ be a D202: Random variable on $P$ for each $j \in J$ such that

(i)	$X = \{ X_j \}_{j \in J}$ is an D2713: Independent random collection on $P$

Let $I \subseteq J$ be a D78: Subset.

Then $\{ X_i \}_{i \in I}$ is an D2713: Independent random collection on $P$.