elementary set theory - Functions and unions on sets

I seek an elementary set theory proof. (All new to me.)

Let $f: X \rightarrow Y$ be a function. $A_i$ are subsets of $X$, $B_i$ are subsets of $Y$.

Prove that $f\left(\bigcup_iA_i\right) = \bigcup_if(A_i)$

I sense that the approach is to show they're subsets of each other, but I can't out how to formulate this.

(Whilst we're at it, is the same true with the intersection?)