All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Your fast, Allah willing, is valid as long as you intended during the night that you would fast to expiate for breaking an oath, and your aforementioned lie does not affect the validity of your fast.
You could have used metaphors, such as saying, “It is an obligatory fast for me,” and so on.
We point out that fasting is not acceptable for the expiation for breaking an oath except for someone who is unable to feed or clothe ten poor people.
Allah knows best.