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

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

Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

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

Uniqueness
Suppose that both $\mathcal S$ and $\mathcal T$ are magmas of sets for $\Phi$ generated by $\mathcal G$.

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


 * $\mathcal S \subseteq \mathcal T$
 * $\mathcal T \subseteq \mathcal S$

By definition of set equality:
 * $\mathcal S = \mathcal T$

Existence
Define $\mathcal S$ by:


 * $\mathcal S := \displaystyle \bigcap \left\{{\mathcal T: \mathcal G \subseteq \mathcal T}\right\}$

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

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

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

By Intersection Preserves Subsets: General Result: Corollary, $\mathcal G \subseteq \mathcal S$.

By Intersection is Subset: General Result, $\mathcal S \subseteq \mathcal T$ for every other magma of sets $\mathcal T$ containing $\mathcal G$.

Thus $\mathcal S$ is a magma of sets for $\Phi$ generated by $\mathcal G$.