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We study the changes in the conformations of brushes upon the addition of crosslinks between the chains using the bond fluctuation model. The Flory-Rehner model applied to uni-axially swollen networks predicts a collapse for large degrees of crosslinking $q$ proportional to $q^{-1/3}$ in disagreement with our simulation data. We show that the height reduction of the brushes is driven by monomer fluctuations in direction perpendicular to the grafting plane and not due to network elasticity. We observe that the impact of crosslinking is different for reactions between monomers of the same or on different chains. If the length reduction of the effective chain length due to both types of reactions is accounted for in a function $\beta(q)$, the height of the brush can be derived from a Flory approach for the equilibrium brush height leading to $H(q)\approx H_{b}\beta(q)^{1/3}$, whereby $H_{b}$ denotes the height of the non-crosslinked brush. more...