For an indexed family $(A_i)_{i \in I}$ the collection of functions $f$ with domain $I$ and $f(i) \in A_i$ for every $i \in I$ is a set

Given: an indexed family $(A_i)_{i \in I}$, and $C := \bigcup_{i \in I} A_i$.

An indexed family with index set $I$ is a function $A$ with $\operatorname{dom} A = I$ (An indexed family $(A_i)_{i \in I}$ is a function with domain $I$; $\{A_i : i \in I\}$ is its range).

$z \in \bigcup_{i \in I} A_i$ holds if and only if $z \in A_i$ for some $i \in I$ ($\bigcup_{i \in I} A_i := \bigcup \{A_i : i \in I\}$, and $\bigcap_{i \in I} A_i := \bigcap \{A_i : i \in I\}$ for $I \neq \varnothing$).

$f \in B^{A}$ holds if and only if $f : A \to B$ (The set $B^{A}$ of all functions $A \to B$).

We write $f : A \to B$, and say $f$ is a function from $A$ to $B$, when $f$ is a function with $\operatorname{dom} f = A$ and $\operatorname{ran} f \subseteq B$ (A function is a relation $f$ with $(a,b) \in f$ and $(a,c) \in f$ implying $b = c$; $f : A \to B$, the value $f(a)$, domain and codomain).

For any parameters $\bar p$ and any set $x$, there is a set $y$ whose elements are exactly the elements $z$ of $x$ for which $\varphi(z,\bar p)$ holds (The Axiom Schema of Separation: for each formula $\varphi$, $\forall \bar p\,\forall x\,\exists y\,\forall z\,(z \in y \leftrightarrow (z \in x \wedge \varphi(z,\bar p)))$).

$x \subseteq y$ means that every element of $x$ is an element of $y$ (Subset $x \subseteq y$, proper subset $x \subsetneq y$, and the separation notation $\{\, z \in x : \varphi(z) \,\}$).

$\operatorname{ran} R := \{\, b : \exists a\ (a,b) \in R \,\}$ (Relation, $\operatorname{dom} R$, $\operatorname{ran} R$, $\operatorname{fld} R$, and the specialisations "relation from $A$ to $B$" and "relation on $A$").