Correspondence Theorem for Ring Epimorphisms/Bijection

Theorem
The direct image mapping $\pi^\to$ and the inverse image mapping $\pi^\gets$ induce reverse bijections between $I$ and $J$, specifically:


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

Proof
The first statements follow from:
 * Preimage of Ideal under Ring Homomorphism is Ideal
 * Image of Ideal under Ring Epimorphism is Ideal

The last statement follows from:
 * Image of Preimage of Ideal under Ring Epimorphism
 * Preimage of Image of Ideal under Ring Homomorphism