There are many exercises I come across where the objective is to prove the existence of a function. Consider the following three examples from various posts on this website:
Let $f:\mathbb N \times \mathbb N \to \mathbb N$ be a function, and let $c$ be a natural number. Show that there exists a function $a : \mathbb N \to \mathbb N$ such that $$a(0) =c $$ and $$a(n++)=f(n,a(n)) \text{ for all } n \in \mathbb N$$
Show that if $X$ and $Y$ are disjoint sets and $f:X\to Y$ and $g:Y\to Z$ are functions, then there exists a unique function $h:X\cup Y\to Z$ such that $h\circ\iota_{X\to X\cup Y}=f$ and $h\circ\iota_{Y\to X\cup Y}=g$.
3) Exercise 3.5.7 of Tao's Analysis I: A question about direct sums.
Let $X,Y$ be sets, and let $\pi_{X\times Y\to X}: X \times Y \to X$ and $\pi_{X\times Y\to Y}: X \times Y \to Y$ be the maps $\pi_{X\times Y\to X}(x,y):=x$ and $\pi_{X\times Y\to Y}(x,y):=y$; these maps are known as the co-ordinate functions on $X \times Y$. Show that for any functions $f: Z \to X$ and $g: Z \to Y$, there exists a unique function $h: Z \to X \times Y$ such that $\pi_{X\times Y\to X} \circ h =f$ and $\pi_{X\times Y\to Y} \circ g =f$.
I recently saw how Cartesian Products, the Power Set Axiom, and the Axiom of Specification could be used to construct a set, denoted $Y^X$, that consists of all functions from set $X$ to set $Y$...where $f \in Y^X \iff \text {$f$ is a function with domain $X$ and range $Y$}$. The proof can be found here: Hints for proving that the collection of all set functions whose domain is the set $X$ and whose range is a subset or equal to the set $Y$ is a set..
Now, referring back to the three aforementioned examples, in all three cases, we already know the domain and codomain.
For the first example, $a \in \mathbb N^{\mathbb N}$. In the second example, $h \in (X \cup Y) ^{Z}$. In the third example, $h \in Z^{X \times Y}$.
Because we know that the set $codomain^{domain}$ exists, and such a set contains all functions that have these corresponding domains and codomains, I fail to see why any proof needs to occur at all.
Regardless of the properties that functions $a$ and $h$ have, they must already exist because if the set that contains them exists, then clearly so do the elements of said set!
Any clarity would be greatly appreciated. Thanks!
If you don't see why a problem is nontrivial, it often helps to consider silly variations of it which are outright wrong.
To start with, consider:
When we write "Show that there exists a function such that [stuff]," what we're proving is that [stuff] is actually a satisfiable constraint. Knowing that the set of all functions with the given domain/codomain exists is irrelevant: we know that if a function with the properties we want exists then it's in that set, but does it actually exist in the first place?
Uniqueness is similarly nontrivial. Consider:
Again, we have a certain property $P$ of functions that we want to understand. The set $\mathbb{N}^\mathbb{N}$ of all functions with the appropriate domain/codomain exists, but that doesn't help answer the question: how many functions $\mathbb{N}\rightarrow\mathbb{N}$ satisfy $P$? (In particular, for existence we need "$\ge 1$" and for uniqueness we need "exactly $1$").