I proved the following two similar theorems about funcoids and reloids:
Theorem
- For a complete funcoid $latex f$ there exist exactly one function
$latex F \in \mathfrak{F}^{\mho}$ such that
$latex f = \bigcup^{\mathsf{FCD}} \left\{ \{ \alpha \} \times^{\mathsf{FCD}} F(\alpha) | \alpha \in \mho \right\}$. - For a co-complete funcoid $latex f$ there exist exactly one function
$latex F \in \mathfrak{F}^{\mho}$ such that
$latex f = \bigcup^{\mathsf{FCD}} \left\{ F(\alpha) \times^{\mathsf{FCD}} \{ \alpha \} | \alpha \in \mho \right\}$.
Theorem
- For a complete reloid $latex f$ there exist exactly one function
$latex F \in \mathfrak{F}^{\mho}$ such that
$latex f = \bigcup^{\mathsf{RLD}} \left\{ \{ \alpha \} \times^{\mathsf{RLD}} F(\alpha) | \alpha \in \mho \right\}$. - For a co-complete reloid $latex f$ there exist exactly one function
$latex F \in \mathfrak{F}^{\mho}$ such that
$latex f = \bigcup^{\mathsf{RLD}} \left\{ F(\alpha) \times^{\mathsf{RLD}} \{ \alpha \} | \alpha \in \mho \right\}$.
See this online article for definitions of used concepts and proofs.