The problem setting is very simple. Suppose we have three variables x, y and z and a constraints C/3 predicate that is satisfied by the three variables C(x,y,z), but C/3 might not be the only constraint x, y and z meet, so the question is how do I formulate the set of all such constraints using logic formula.
Can I say
it feels awkward because the quantifier is supposed to restrict the variable instead of the predicate in a logic formula so the question is how should I approach this formulation, thanks.
If I understood your question correctly, given three variables $x,y,z$ and a set of constrains $R$, you want to express the set $S$ of all constrains $C \in R$ (were $C$ must be some kind of equation) such that $(x,y,z)$ satisfy $C$, and we will denote this by $C(x,y,z)$. This set would be defined as (you were almost correct):
$S = \{C \in R | C(x,y,z) \}$. This is the expression you are looking for.
However, if you want the set $V$ of variables $x,y,z$ that satisfy the constrain $C$ but may also satisfy other constrains, this would be expressed as:
$V = \{(x,y,z) \in \mathbb{R} | C(x,y,z), C \in A\}$.
Where $A \subseteq R$ would be the set of constrains that $(x,y,z)$ satisfy.
I am a little confused, since $S$ is a set of constrains but $V$ is a set of points (e.g. variables), but you want a set of constrains and that the variables are restricted somehow. But, if you want a set of constrains, the variables play a little role, mainly only being used as the domain of the constrains. I hope this is useful.