Reading a first order logic definition, with for all I'm trying to read the following FOL for a functional binary relation:
$R \subset X \times Y$
$\forall x \in X, \forall y \in Y, \forall z \in Y, (x,y) \in R \land (x,z) \in  R \implies y = z$
I'm confused on how to read this.  It's my understanding  "∀ __ is True if the value of __ is True for all values of x" and that "∀ __ is False if the value of __ is False for any value of x.
So above, $\forall z \in Y$ will bind $z$ to values where $y \ne z$ and the entire statement will be false.  Am I reading this wrong?

You are misunderstanding the bracketing here:
$\forall x \in X, \forall y \in Y, \forall z \in Y, (x,y) \in R \land (x,z) \in  R \implies y = z$
Should be:
$\forall x \in X, \forall y \in Y, \forall z \in Y, ((x,y) \in R \land (x,z) \in  R) \implies y = z$
