Inverse Image under Embedding of Image under Relation of Image of Point

Theorem
Let $S$ and $T$ be sets.

Let $\mathcal R_S$ and $\mathcal R_t$ be relations on $S$ and $T$, respectively.

Let $\phi: S \to T$ be a mapping with the property that:


 * $\forall p, q \in S: \left({ p \mathrel{\mathcal R_S} q \iff \phi(p) \mathrel{\mathcal R_T} \phi(q) }\right)$

Then for each $p \in S$:


 * $\mathcal R_S (p) = \phi^{-1}\left({\mathcal R_T \left({ \phi(p) }\right) }\right)$

Proof
Let $p \in S$.

Thus by the Axiom of Extension:


 * $\mathcal R_S (p) = \phi^{-1}\left({\mathcal R_T \left({ \phi(p) }\right) }\right)$