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

Proof
Follows from Correspondence Theorem for Ring Epimorphisms/Bijection