Deep neural network (DNN) has been increasingly deployed as a key component in safety-critical systems. However, the credibility of DNN components is uncertain due to the absence of formal specifications for their data preconditions, essential for ensuring trustworthy postconditions. In this paper, we propose a guess-and-check-based framework PreBoxes to automatically synthesize Boxes sufficient preconditions for DNN concerning rich safety and robustness postconditions. The guess phase effectively provides potentially non-trivial candidate sets and continuously reduces overhead throughout all designed heuristics. The check phase provides a provable guarantee. The entire framework supports automatic and adaptive iterative running to obtain weaker preconditions as well. Such resulting preconditions can be leveraged to shield DNN for safety and enhance the interpretability of DNN in application. PreBoxes has been evaluated on over 20 models with 23 trustworthy properties of 4 benchmarks and compared with 3 existing typical schemes. The results show that not only does PreBoxes generally infer weaker non-trivial sufficient preconditions for DNN than others, but also it expands competitive capabilities to deal with both complex properties and Non-ReLU complex structured networks.