Is my prayer valid if I intended to pray Fajr but then recalled in my heart that it was the Friday prayer by mistake, and is what Ibn Uthaymeen said considered to make my prayer valid?

If a worshipper makes a mistake in their intention and intends the Friday prayer instead of the Fajr prayer, then the Fajr prayer is invalid and must be repeated; because specifying the prayer by intention is a condition for its validity.

Shaykh Ibn Uthaymeen mentioned in "Ash-Sharh Al-Mumti'" that the worshipper must intend the specific prayer. So, if he wants to pray Dhuhr, he must intend the Dhuhr prayer.

As for what Ibn Uthaymeen stated regarding the validity of the prayer if a long period elapses between the intention and the act, without revoking the intention, this differs from the case of a complete change of intention.