I'm trying to understand AEP(Asymptotic Equipartition Property) theorem but I couldn't understand following result:
Let $X_1, X_2, \dots , X_n$ be a sequence of i.i.d. random variables which take on the values from finite alphabet $\mathcal{X}$ and $X_i\sim P_X$. Prove that $$P_X(X_1,X_2, \dots , X_n) = P_X(X_1)P_X(X_2)\dots P_X(X_n)$$
We know in general that functions of independent RVs are also independent RVs but how does this help us to prove the result? Also I don't understand exactly the meaning of $$P_X(X_1,X_2, \dots , X_n) \tag{1}$$ By definition $$P_{X^n}(x_1 , x_2, \dots , x_n) = \mathbb{P}(X_1 = x_1,X_2 = x_2, \dots , X_n = x_n)$$but $(1)$ is really confusing because arguments are RVs instead of certain values and it seems self-referential.