Notation for indicating a multi-parameter function is bijectivewhen all but one parameter is held constant

Question

Suppose I have a function that takes two or more values, such as $f(x, y)$ or $g(a, b, c)$. When all but one particular parameter's value becomes fixed, the relationship between the unfixed parameter and the function's output becomes one-to-one. Using the same sort of notation that incorporates $\forall$ and $\exists$ (I think it is called first-order logic), how can I indicate that $f$ and $g$ have this property? I am guessing fragments of such a notation for $f$ might be $\forall{t,v}$, $v = f(t, u)$, and $\exists{u}$.

Answer 1

If $f$ takes $n$ arguments, then a formula for being one-to-one in each is: $$\bigwedge_i\big(\forall x_1,\dots, x_n, y:\ f(x_1,\dots, x_i, \dots, x_n) =f(x_1,\dots, y, \dots, x_n) \, \implies\, x_i=y \big) $$