I am trying to understand the proof of the following theorem:
Let $f \in \mathbb{C}[x,y]$ and $f = \prod_{i=1}^{r} f_i^{n_i}$ be a decomposition of $f$ into irreducible factors. Then, $V(f) = \cup_i V(f_i)$ is the decomposition of $V(f)$ into irreducible components, and $I(V(f)) = (f_1...f_r)$.
Pf. (of second statement)
$I(V(f)) = I(\cup V(f_i)) = \cap_i I(V(f_i)) = \cap_i(f_i) = (f_1...f_r).$
I have trouble seeing why the second last equation holds. Please can you explain why that equation holds?