Prove f(indexed intersection Ai) is a subset of indexed intersection Ai.

Question

Here is what I'm working on:

I started with:

I'd love to see how to finish the proof. Thank you for your time.

Answer 1

Hint. You have:

Let $y \in f(\bigcap_i A_i)$, then there is $p \in \bigcap_i A_i$ such that $f(p) = y$. Then forall $i$, $p \in A_i$.

If $p \in A_i$, then - by definition - $y = f(p) \in f(A_i)$, hence $y \in f(A_i)$ for all $i$.