Correspondence Theorem for Quotient Rings/Bijection

Theorem
The direct image mapping $\pi^\to$ and the inverse image mapping $\pi^\gets$ induce reverse bijections between the ideals of $A$ containing $\mathfrak a$ and the ideals of $A/\mathfrak a$, specifically:

Let $I$ be the set of ideals of $A$ containing $\mathfrak a$.

Let $J$ be the set of ideals of $A/\mathfrak a$.

Then:
 * 1) For every ideal $\mathfrak b \in I$, its image $\pi^{\to}(\mathfrak b) = \pi(\mathfrak b) \in J$.
 * 2) For every ideal $\mathfrak c \in J$, its preimage $\pi^{\gets}(\mathfrak c) = \pi^{-1}(\mathfrak c) \in I$.
 * 3) The restrictions $\pi^\to : I \to J$ and $\pi^\gets : J \to I$ are reverse bijections.