Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets

Theorem
Let $X$ be a set, and let $\Phi := \set {\phi_i: i \in I}$ be a collection of partial mappings with codomain $\powerset X$, the power set of $X$.

Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.

Then the magma of sets generated by $\GG$ exists and is unique.

Uniqueness
Suppose that both $\SS$ and $\TT$ are magmas of sets for $\Phi$ generated by $\GG$.

Applying condition $(2)$ for these twice, we obtain:


 * $\SS \subseteq \TT$
 * $\TT \subseteq \SS$

By definition of set equality:
 * $\SS = \TT$

Existence
Define $\SS$ by:


 * $\ds \SS := \bigcap \set {\TT: \GG \subseteq \TT}$

where $\TT$ ranges over the magmas of sets for $\Phi$ on $X$.

From Power Set is Magma of Sets, there intersection is at least one such $\TT$.

From Intersection of Magmas of Sets is Magma of Sets, $\SS$ is a magma of sets for $\Phi$.

By Intersection Preserves Subsets: General Result: Corollary, $\GG \subseteq \SS$.

By Intersection is Subset: General Result, $\SS \subseteq \TT$ for every other magma of sets $\TT$ containing $\GG$.

Thus $\SS$ is a magma of sets for $\Phi$ generated by $\GG$.