Definition:Supremum of Set/Real Numbers/Propositional Function

Definition
Let $\family {a_j}_{j \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$.

Let $\map R j$ be a propositional function of $j \in I$.

Then we can define the supremum of $\family {a_j}_{j \mathop \in I}$ as:


 * $\displaystyle \sup_{\map R j} a_j := \text{ the supremum of all $a_j$ such that $\map R j$ holds}$

If more than one propositional function is written under the supremum sign, they must all hold.