Let $X$ and $Y$ be arbitrary sets and let $f:X\times Y\to Y$ be some well-defined function.
Then, function $f:X\times Y\to Y$ satisfies Property P if for all $y,y'\in Y$, there exists some $x\in X$ such that $f(x,y)=y'$.
Suppose there exists $y^*\in Y$ for which there exists no $x\in X$ such that $f(x,y^*)=y^*$. Then, is Property P violated?
I guess that another way to frame my doubt is this: in the expression "$\forall y,y'\in Y$", can $y=y'$ or must it be the case that $y\neq y'$?
$\forall y, y'\in Y$ means you are free to choose any two elements from $Y$ , doesn't matter whether they are distinct or not.
As you have written it, Property P requires that if $y$ and $y'$ are elements of $Y$ (not necessarily unique), then there must be an element $x$ of $X$ such that $$f(x,y)=y'.$$ Both $y^*$ and $y^*$ are elements of $Y$, hence if there is no $x$ in $X$ such that $$ f(x,y^*) = y^*, $$ then $f$ fails to satisfy Property P.
This is a mathematical meaningful part, an expression not more. There is a lot of need for other expressions to make this a unique expression with only one meaning. There is need for what $y, y'$ should be. The prime might be a distinction of every type. The expression contains a comma. The most unique situation is therefore decimal separation. In grammars this may to separate part of a sentence. In maths sentences are very different and fundamental statements. The expression may be part of a sentence in either meaning. It is possible to consider the comma as a separator of parataxis, asyndetic raw and so on.
In math it is possible to make clear that variables are different as mentioned by lost-in-space but it also possible that the use of the variable is different like in the second answer for example a derivation. That must not necessarily be a derivation in the sense of Newton or Leibniz. In modern set theory there is a distinction between two type of variable use.
One example may be $y$ is a parameter and $y'$ is a variable. That can be driven further. The use as examples or samples for example.
The only more standard difference is that between $y$ and $Y$ in pure mathematics. $Y$ is a symbol a more advanced object, set, class, group, ring, module, algebra. It can even denote a function or relation between there or others. $y$ is that a subset, element, representation, sample, example, variable, parameter and so on of $Y$.
The very same with $y'$.
There is of course as well a standard as a norm for ∈.
The same for the all, universal quantor, quantifier ∀.
There is a standard and a norm that these only act on the very next symbol on the right of it.
These are well documented on ISO_31-11. Form that reference the given expression is partly in the table for logic:
∀ ∀x∈A p(x) (∀x∈A) p(x) universal quantifier for every x belonging to A, the proposition p(x) is true The "∈A" can be dropped where A is clear from context.
So this allows for one symbol after the universal quantifier to drop the element sign.
But in your expression there is no symbol after that dropped sign there is comma and element of sign after another symbol, letter.
The reference is a good starting point for notation and symbols in mathematics as well.
So caused by the fact the there is given an expression and not a valid statement either of a sentence, a lemma, a corollary, a proof, an example, a calculation, a definition of anything else it remains a free expression and there is no information in it what logical connective or other type of connective holds between $y, y'$ it is a loose as a by naive set theory of pure mathematics.
So the best answer is contained in Asyndeton. By accident this is not a pure mathematics texts and contains misnomers. This reference give some more clues about what is missing: Conjunction_(grammar). You will agree with me that this is not a Polysyndeton.
By analysing a sentence and relate it the categories of what sentence this might be the flaws in real language are even more evident but the text will grow to long.
Mathematics and even formal mathematics is poor in variability compared to real language. There is a bunch of initiative for example CMI for Comprehensive Mathematics Instruction already for instructors of primary and secondary maths to make the instructions of Mathematics easier. This reference names some of the basics competences concerned. Some extend this to the demand for readable Mathematics in publications.
To translate this there are some sources that may help. There are CAS in with the symbol are allowed to by defined and the signs defined and a atomic definition with example presented, for example Maple, Mathematica. There are solvers like COQ or LEAN that pose the same but may better check for plausiblity or consistency. There is for example this ways-help-students-understand-math-matthew-beyranevand.
The pathway is introduction to literacy in mathematics if you want to go further. This should be a start for the person who gave this expression to you as well. Have fun.
