Is the converse of this implication correct?

Question

Is the converse of this implication correct? 1

Where $\{A_i : i\in I\}$ is the indexed family of subsets of $Y$ and $A$ is a subset of $Y$.

If the converse of the implication 1 is not always true, then where did I went wrong? 2

Answer 1

If $\bigcup_{i \in I} A_i \subseteq A$ then indeed for each (fixed) $i$, $$A_i \subseteq \bigcup_{i \in I} A_i \subseteq A \text{ so, in conclusion: } \forall i \in I: A_i \subseteq A$$

and the reverse implication is also true.