Take a decision problem $Q$, which maps encoded instances of a problem, i.e., $\lbrace 0, 1 \rbrace \ast$ to the solution set $\lbrace 0, 1 \rbrace$.
Since $Q$ is in $NP$, there exists a verification algorithm $A$, which takes an input and a certificate, in that order, as arguments and maps those to $\lbrace 0, 1 \rbrace$, depending on whether there exists a certificate proving that the solution to the input is $1$.
If this presentation of what a verification algorithm does is correct, then it is unclear to me if, for a binary-encoded problem instance $x$,
$$Q(x) = 1 \iff \exists y (A(x, y) = 1).$$
It seems to me that if there is a certificate, this means that the decision for that input is affirmative, so that in theory, you could attach sufficient information to a certificate such that it characterizes a solution to the decision problem, iterate through the entire set of certificates and, once you have found one that satisfies $A(x, y) = 1$, you have also found a solution to the problem for the respective input.
Maybe someone could explain to me if that equivalency above is wrong and if so, why. I know the difference between a problem and the verification part, but then again, on some level I do not.
Neither am I sure about anything else, so feel free to point out mistakes.