Extreme events (such as earthquakes, hurricanes, etc.) pose a dual challenge to the reliability and serviceability of IoT systems. With regard to this challenge, by publishing some tasks and then encouraging the public to assist in real-time data collection through their mobile terminals (namely, the mobile crowdsourcing-assisted IoT systems), is expected to play an important role in secondary disaster prevention and personnel rescue in extreme events. However, it has weaknesses in terms of security, flexibility, and efficiency. As an elegant solution, identity-based broadcast proxy re-encryption (PR-IBBE) enables flexible access authorization sharing and efficient broadcast distribution of encrypted tasks via the cloud. However, their security relies on fully-trusted or semi-trusted cloud assumptions, which are hard to be implemented in real-world scenarios. And the cloud is more vulnerable in an emergency event since there is a lack of effective management. Motivated by that, we propose the verifiable PR-IBBE (VPR-IBBE) scheme, which realizes a cross-domain identity-based broadcast task file secure authorization access, and empowers the verifiability and reputability of re-encrypted ciphertext under the untrusted cloud setting. This mechanism ensures that the relevance between the re-encrypted ciphertext and the original ciphertext can be publically verified, so the cloud can defend itself if there is a malicious accusation of forging the re-encrypted ciphertext. Through rigorous formal security proofs, we demonstrate that VPR-IBBE attains the indistinguishability of ciphertext against selective identity chosen ciphertext attack (IND-sID-CPA), and is also resistant to the collusion attack between the untrusted cloud and the cooperative performer. Theoretical comparison and experimental results demonstrate the practicability of our VPR-IBBE scheme, as well as the superiority over representative related works.