About tahajjud prayer when to be performed

Yesterday I had gone to sleep late due to my studies at like 1 am to 2 am ish. So before going to sleep I prayed Tahajjud and Witr prayer so that when I wake up I wouldn’t have to rush my suhoor and eat it peacefully. But after telling my mother this, she told me that tahajjud is to be strictly prayed after waking up. Is it true? If it is then can you show me the proofs? جزاكم الله خيرا كثيرا in advance.