$(A\cap B)\cup C = A \cap (B\cup C)$ if and only if $C \subset A$ I have a set identity: $(A \cap B) \cup C = A \cap (B \cup C)$ if and only if $C \subset A$.
I started with Venn diagrams and here is the result:

It is evident that set identity is correct. So I started to prove it algebraic:
1) According to distributive law: $(A \cap B) \cup C = (A \cup C) \cap (B \cup C)$
2) ...
I stuck a little. Because $C$ is a subset of $A$. I thought of pulling out: $(B \cup C)$ but it seems wrong step to me.
How to prove this identity having in mind that $C \subset A$?
Updated
Venn diagram for $C ⊈ A$


By Distributivity, 
$$ (A \cap B) \cup C = (A \cup C) \cap (B \cup C ) $$
Suppose $ C \subset A $, then $A \cup C = A $ by definition. Therefore $ (A \cap B) \cup C = A \cap (B \cup C )$
Conversely, suppose $ (A \cup C) \cap (B \cup C ) = A \cap (B \cup C )$. We show $C \subset A $. But, by the equality, it is obvious that $ A \cup C = A $. In other words, $C $ must be inside $A$
