Given a collection of functions $f_i$ with the same domain, how to replace with values (w/o axiom replacement)

Question

I know from a collection of ordered pairs we can project onto the first coordinate. I'm interested if there's a way (without using the axiom of replacement) to "project" a collection of functions onto values at a particular point in the domain. For simplicity, assume the collection of functions all have the same nonempty domain (axiom of choice is legal).

Thanks for any insight!

Answer 1

My set theory is a bit rusty, so please check the reasoning carefully.

1. I'm assuming you define an ordered pair as $(x, y) = \{\{x\}, \{x, y\}\}$. I'm assuming that a function is its graph, i.e. a set of ordered pairs, with no additional information about the domain and codomain. I'm also assuming that by "collection" you mean "set".
2. Let $F$ be your set of functions. Let $S = \cup F$ be the union of all the functions in your collection. $S$ is a set of ordered pairs. Here we only use the axiom of union.
3. Given this set of ordered pairs $S$, you can get a set $M$ such that $S \subseteq M \times M$. I think that $M = \cup \cup S$ should do. Again, this only requires the axiom of union.
4. Now, $M$ contains all the domains and all the ranges of your functions. Thus, the set that you're looking for is a subset of $M$, so it can be built using the axiom schema of specification.

And that is it. No choice, no replacement, and we don't really use the condition that all the domains are the same.

I also think that step 3 doesn't really change much if you use another definition of ordered pairs.