Definitions are needed to define objects and such, however I am confused as to where definitions come from. I feel that they cannot be something that we arbitrarily define because simply saying something exists does not ensure its existence and further defining something from nothing may eventually lead to a contradiction. So it seems a strange situation that we need definitions to have something to work with and prove stuff about, and yet these definitions surely need to go thru a process of proof themselves. Can someone explain what definitions are in the most primitive sense and where they come from?
Edit: let me elaborate on my confusion, I have been reading up on different logic books and somewhere came upon a definition of proof as
"Let T be a set of first order formulas. A proof from (proper axioms) T is a finite sequence of formulas ("steps") such that every step is either a logical axiom, a member of T, or the result of applying a rule of inference to previous steps in the proof."
-a proper axiom set is for example set theory axioms.
Now from this way of defining proof, it seems like the only definitions that can be used are those of the proper axioms.
As per Henning Makholm's request, lets talk about the definition of a function. Where does the $(\forall x(x\in A\implies\exists! y((x,y)\in f))$ come from? in regards to the definition just given of proof, it is not a logical axiom nor a proper axiom. Is this something that is always assumed and therefore added to a premise in any proof?
Edit2: I think Alfred yerger answered this best in his comment.I'm not saying anyone is wrong I just got the connection after reading his comment. This makes most sense when I think about it in the context of the definition of proof I provided. Definitions are shorthand for concepts we wish to define and prove about and this comes in 2 possible ways, in the hypothesis (e.g. 'let f be a function such that...') or is the property which we try and prove some particular object has (e.g. 'show f=... is a function') in this way, the definition does not claim something exists. That happens during the hypothesis of a proof as was noted by yerger. Sorry if I was being too vague when first posing this question. I will award the answer to someone who could maybe flesh this answer out a little more and maybe provide some good special cases or instances of ways definitions come about being defined. Thank you all for the contributions and I really wish I could award more than 1 point.