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

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

Let $\RR_S$ and $\RR_t$ be relations on $S$ and $T$, respectively.

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


 * $\forall p, q \in S: \paren {p \mathrel {\RR_S} q \iff \map \phi p \mathrel {\RR_T} \map \phi q}$

Then for each $p \in S$:


 * $\map {\RR_S} p = \phi^{-1} \sqbrk {\map {\RR_T} {\map \phi p} }$

Proof
Let $p \in S$.

Thus by definition of set equality:


 * $\map {\RR_S} p = \phi^{-1} \sqbrk {\map {\RR_T} {\map \phi p} }$