I'm trying to understand the Cook-Levin theorem proof, as it attempts to create a polynomial-time reduction from any NP
problem to SAT
(as presented in the book by Michael Sipser).
Most requirements are absolutely clear, but I don't understand why the $\phi_{cell}$ formula is required.
Of course, a tableau with improper values, such that fails $\phi_{cell}$, is broken and therefore will not "translate" correctly to the original NP problem on the other side of the reduction. We can easily "break" a tablaue by putting a $q_{accept}$ somewhere where it shouldn't, and the machine's language will change.
But I seem to miss something basic. A reduction says that if A reduces to B, then for each $w\in\Sigma^* $, $w\in A$ iff $f(w)\in B$.
The $f(w)$ part may be the "broken" tableau, but can we really call it $f(w)$? After all, such broken tableaus are never in the image of $f$.
So, why do we care about them?