Intending to fast in Rajab, then bathing from a wet dream upon waking and completing the fast: Is the fast valid?

Question:
May Allah reward you with good and do you good. He says: In the month of Rajab, I intended to fast on one of the days. When I woke up, I found that I had had a wet dream, and that was a considerable time after the Fajr adhan. I was lazy about getting up to bathe and pray, so I continued sleeping until morning. When I woke up, I bathed and then prayed. What is the ruling on my fast, and is there any expiation required of me, or not? We ask for clarification and guidance. May Allah reward you with good.

Answer:
First, singling out Rajab, or singling out a part of it, for fasting has no basis and is considered a bid'ah (religious innovation). However, if it is your custom to fast on Mondays and Thursdays, or to fast three days of every month, and you wish to fast this fast in the month of Rajab, there is no harm in that. Rajab is like the other months regarding fasting on Mondays and Thursdays, or fasting three days from it.
As for what you mentioned, that you had a wet dream and did not bathe until after the dawn had appeared, and you intended to fast before that, there is no harm in that, and the fast is valid. It is valid for a person in a state of major ritual impurity to intend to fast and abstain, then bathe afterwards, because ritual purity is not a condition for fasting; rather, ritual purity is a condition for prayer. Yes.