Equality of two sets (using Boolean algebra)

Hello. I recently started my course, so this is pretty trivial.

Prove:

$\displaystyle \displaystyle \[f\left( {\bigcup\limits_{i \in \ell } {{A_i}} } \right) = \bigcup\limits_{i \in \ell } {f\left( {{A_i}} \right)} \]$

Well, analogous examples demonstrated in my university begin with

$\displaystyle \displaystyle \[\forall y \in f\left( {\bigcup\limits_{i \in \ell } {{A_i}} } \right) \Rightarrow \left( {\exists x:f\left( x \right) = y} \right) \wedge \left( {x \in \bigcup\limits_{i \in \ell } {{A_i}} } \right)\]$

The problem is that I am not sure how to represent this highly abstract union of sets since

$\displaystyle \displaystyle \[x \in \bigcup\limits_{i \in \ell } {{A_i}} \Rightarrow x \in {A_i} \wedge i \in \ell \]$

I guess wouldn't make enough sense (I could tell the same about intersection of those very same sets).

So what is missing on my mind? (Worried)

Thanks for help.