He had a wet dream and could not do ghusl so he did wudoo’ and prayed

I am a disabled man sitting on wheelchair. Alhamdulillah.
I had a wet dream in some night before Fajr prayer. Due to the weakness of my body and strength I could not make ghusl nor could I change my clothes except the next morning. I did not want to miss Fajr prayer on time, so I made ablution like I normally do for prayers, and then prayed. Is that permissible? If it is not then what shall I do?

Praise be to Allaah.

Firstly: 

We ask Allaah to grant you well being in this world and in the Hereafter. 

Anyone who becomes junub as the result of a wet dream or intercourse has to do ghusl, because Allaah says (interpretation of the meaning): 

“O you who believe! When you intend to offer As-Salaah (the prayer), wash your faces and your hands (forearms) up to the elbows, rub (by passing wet hands over) your heads, and (wash) your feet up to the ankles.  If you are in a state of Janaaba (i.e. after a sexual discharge), purify yourselves (bathe your whole body)”

[al-Maa’idah 5:6] 

If he cannot do ghusl because there is no water available, or because he is not able to use it, then he should do tayammum and pray, because Allaah says (interpretation of the meaning): 

“But if you are ill or on a journey, or any of you comes after answering the call of nature, or you have been in contact with women (i.e. sexual intercourse), and you find no water, then perform Tayammum with clean earth and rub therewith your faces and hands.  Allaah does not want to place you in difficulty, but He wants to purify you, and to complete His Favour to you that you may be thankful”

[al-Maa’idah 5:6]. 

Similarly, if a person has a wet dream and fears that he may become sick if he uses cold water, and there is no way of heating it, or the water is far away from him and he cannot reach it, and there is no one to bring it to him, then he may do tayammum and pray. 

Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (1/151): 

If a person is sick and unable to move, and there is no one who can bring him water, then he is like one who has no water, because there is no way he can get to the water. … If he has someone who can bring him water before the time for prayer ends, then he is like one who has water available. End quote. 

The sick person who cannot wash all of his body, but is able to wash part of it, must wash whatever he can then do tayammum for the rest, because Allaah says (interpretation of the meaning): 

“So keep your duty to Allaah and fear Him as much as you can”

[al-Taghaabun 64:16] 

Based on this, you have to repeat that prayer. 

Secondly: 

Semen is pure (taahir) according to the more correct of the two scholarly opinions, hence you do not have to wash your clothes or change them. See question no.2458. 

And Allaah knows best.