Change of Index Variable of Supremum

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Theorem

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

Let $R \left({i}\right)$ be a propositional functions of $i \in I$.


Let $\displaystyle \sup_{R \left({i}\right)} a_i$ be the indexed supremum on $\left \langle {a_i} \right \rangle$.


Then:

$\displaystyle \sup_{R \left({i}\right)} a_i = \sup_{R \left({j}\right)} a_j$


Proof


Sources