A circular disc impacting on a water surface creates a remarkably vigorous jet. Upon impact an axisymmetric air cavity forms and eventually pinches off in a single point halfway down the cavity. Immediately after closure two fast sharp-pointed jets are observed shooting up- and downwards from the closure location, which by then has turned into a stagnation point surrounded by a locally hyperbolic flow pattern. This flow, however, is {it not} the mechanism feeding the two jets. Using high-speed imaging and numerical simulations we show that jetting is fed by the local flow around the base of the jet, which is forced by the colliding cavity walls. Based on this insight, we then show how the analytical description of a collapsing void (using a line of sinks along the axis of symmetry) can be continued beyond the time of pinch-off to obtain a quantitative model for jet formation which is in good agreement with the numerical and experimental data.