# Meaning of the statement $\forall x \exists y \forall z (z \neq y \iff f(x) \neq z)$

I need to understand the logic of below FOL statement. Can someone help?

$$\forall x \exists y \forall z (z \neq y \iff f(x) \neq z)$$

Does this imply that x, y and z cannot be same or f(x) has no value?

• Predicate logic questions might be better suited in the computer sceince SE. – solarflare Apr 15 '19 at 5:42

"for all $$x$$, there exists a value of $$y$$ such that for all $$z$$,
$$z\neq y$$ if and only if $$z \neq f(x)$$".
This can be simplified: \begin{align} & & \forall x \exists y \forall z (z\neq y \iff z \neq f(x))\\ &\implies & \forall x \exists y \forall z (z=y \iff z = f(x))\\ &\implies & \forall x \exists y \forall z (y = f(x))\\ &\implies & \forall x \exists y (y = f(x))\\ \end{align}
If we denote the set of all values of $$x$$ by $$X$$ and the set of all values of $$y$$ by $$Y$$, then this tells us that the function $$f$$ maps every $$x$$ in $$X$$ to a $$y$$ in $$Y$$. That is, $$f: X \to Y$$.