Change of Index Variable of Supremum

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Theorem

Let $\family {a_i}{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$.

Let $\map R i$ be a propositional functions of $i \in I$.


Let $\displaystyle \sup_{\map R i} a_i$ be the indexed supremum on $\family {a_i}$.


Then:

$\displaystyle \sup_{\map R i} a_i = \sup_{\map R j} a_j$


Proof


Sources