I'm trying to piecewise define a function $h$ using two other functions $f$ and $g$. I want to use $h$ to draw conclusions on a certain set $T$ that's a union of two other sets $A$ & $B$.
$ h(n_z) = \begin{cases} f(n_z), & \mbox{if } f(n_z) \in A \wedge f(n_z) \notin B \wedge g(n_z) \notin A \\ g(n_z), & \mbox{if } g(n_z) \in B \wedge g(n_z) \notin A \wedge f(n_z) \notin B \\ f(n_z), & \mbox{if } f(n_z) \in A \wedge \exists n_y \in \mathbb N, g(n_y) \in B \end{cases} $
My question is, is it legal to do such a definition? Keep in mind that I've already defined $f$ & $g$ in the proof I'm doing and I'm not simply pulling them out of thin air.
Such an definition is legal, or - how it's usually called - well-defined, if
In other words, for each possible state, exactly one of the conditions must be fulfilled.
It is difficult to evalute this in this case, as we don't know the context. It seems, that the case $f(n_z)\notin A, g(n_z)\notin B$ is not considered, but maybe this case does not occur?