I tried finding stuff online but I couldn't about partial matchings.
Definition we're given: A partial matching of $\mathcal{A}$ is a matching of some subfamily $\mathcal{B}\subseteq \mathcal{A}$.
A matching is defined:
A matching of $\mathcal{A}$ is a set of distinct elements $\{a_i\}_{i\in I}$ with $a_i \in A_i$ for all $i\in I$ where $\mathcal{A} = \{A_i\}_{i\in I}$.
Let $\mathcal{I}(\mathcal{A})$ be a family of partial matchings of $\mathcal{A}$.
Prove that if $A \in\mathcal{I}(\mathcal{A})$ and $B\subseteq A$, then $B\in\mathcal{I}(\mathcal{A})$.
I'm not sure where to start really and I tried using Hall's theorem for it to no avail.