Is it permissible to complete the fast of the day if I became pure after Fajr and bathed after Dhuhr, knowing that the intention to fast was made from the night, and is it a condition to bathe before Fajr or to attain purity before Fajr to complete the fast?

If you purify yourself from menstruation before dawn (Fajr) and intend to fast, your fast is valid, even if you do not perform ghusl (definition) (ritual bath) until after that. However, delaying ghusl until noon (Dhuhr) when it is possible to perform it earlier is a grave sin due to missing the Fajr prayer. You must repent and make up the prayer. But if you do not purify yourself until after dawn, fasting on that day is not valid due to the presence of menstruation, for "whenever menstruation is present during any part of the day, the fast of that day is invalidated."