Set of Sets can be Defined as Family

Theorem
Let $\Bbb S$ be a set of sets.

Then $\Bbb S$ can be defined as an indexed family of sets.

Proof
Let $S: \Bbb S \to \Bbb S$ denote the identity mapping on $\Bbb S$:
 * $\forall i \in \Bbb S: S_i = i$

where we use $S_i$ to mean the image of $i$ under $S$:
 * $S_i := \map S i$

Then we can consider $S$ as an indexing function from $\Bbb S$ to $\Bbb S$.

Hence in this case $\Bbb S$ is at the same time both:
 * an indexing set

and:
 * the set indexed by itself.

It follows that each of the sets $i \in \Bbb S$ is both:
 * an index

and:
 * a term $S_i$ of the family of elements of $\Bbb S$ indexed by $\Bbb S$.

Thus we would write $\Bbb S$ as:


 * $\family {S_i}_{i \mathop \in \Bbb S}$