Sets and Functions Defined Componentwise on Families of Vector- and Matrix-Valued Functions

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $p,q\in\mathbb{N}$ and let $S$ be a set. For a function $f:S\to\mathbb{R}^{q}$ and $k\in[q]$, $f_{k}:S\to\mathbb{R}$ is its $k$-th component function; for a function $F:S\to\mathcal{M}_{p\times q}(\mathbb{R})$, $i\in[p]$ and $j\in[q]$, $F_{ij}:S\to\mathbb{R}$ is its entry function in row $i$ and column $j$. For sets $J$ and $B$, $B^{J}$ is the set of functions from $J$ to $B$; for a function $G$ on $[p]\times[q]$, $G(i,j)$ means $G((i,j))$.

(i) Componentwise subset, vector case. Let $D$ be a set of functions from $S$ to $\mathbb{R}^{q}$ and let $A$ be a set. There is exactly one set $Z$ whose elements are exactly the $f\in D$ with

$$f_{k}\in A\qquad\text{for every }k\in[q].$$

(ii) Componentwise function, vector case. Let $D$, $A$ and $Z$ be as in (i), let $B$ be a set and let $\Phi:A\to B$ be a function. There is exactly one function $\Psi:Z\to B^{[q]}$ with

$$\Psi(f)(k)=\Phi(f_{k})\qquad(f\in Z,\ k\in[q]).$$

(iii) Componentwise subset, matrix case. Let $D$ be a set of functions from $S$ to $\mathcal{M}_{p\times q}(\mathbb{R})$ and let $A$ be a set. There is exactly one set $Z$ whose elements are exactly the $F\in D$ with

$$F_{ij}\in A\qquad\text{for all }i\in[p]\text{ and }j\in[q].$$

(iv) Componentwise function, matrix case. Let $D$, $A$ and $Z$ be as in (iii), let $B$ be a set and let $\Phi:A\to B$ be a function. There is exactly one function $\Psi:Z\to B^{[p]\times[q]}$ with

$$\Psi(F)(i,j)=\Phi(F_{ij})\qquad(F\in Z,\ i\in[p],\ j\in[q]).$$