Item 8 (00S1) — The Clowder Project

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Interaction With Weak Inverse Images II. Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation from $A$ to $B$.
1. If $R$ is a total relation, then we have an inclusion of sets
  \[ R_{-1}\webleft (V\webright ) \subset R^{-1}\webleft (V\webright ) \]
  
  natural in $V\in \mathcal{P}\webleft (B\webright )$.
2. If $R$ is total and functional, then the above inclusion is in fact an equality.
3. Conversely, if we have $R_{-1}=R^{-1}$, then $R$ is total and functional.

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