The other evening while searching the internet for random video game related stuff I came across a guide with a program called ps3xport. It’s a program to transfer PSP ISO’s to an official firmware PS3 no need for hacking or modding on the PS3 side.
Now this method is a bit labor intensive and time consuming but worth while. Sky Blade Cloud decided to put a few tutorials up to show how this works. 
The method involves you obtaining your PS3’s IDPS, user number, and HDD backup. Which can be done with the files that will be available at the end of this post.
To acquire this you need to run a program called StealIDPS from a backup of your PS3 HDD. Now I will tell you this, it’s going to be a very time-consuming process but the benefits of this will be awesome. These two guide video posts were made by Sky blade cloud, you may remember him as the man who did the translation for final fantasy type 0 and has done many other translations in the past. The 2 tutorials that he has been kind enough to share with us, walk’s us through steps needed to do this properly. I will be showing you a few photos below the differences between the PSP, PSTV, and the PS3. You will see what the graphical differences are and also see how nice it can be to run PSP games on our official firmware PS3. 
PSP
PSTV
PS3
Full screenshot of the PS3 playing a PSP game on OFW.
Link for tools used in part (1) https://drive.google.com/file/d/0B0cAQvI4fv9GRnZqaHNvcUtwUFk/view
Link for tools in part (2) https://drive.google.com/file/d/0B0cAQvI4fv9GWU9NNzdxWjBLa3M/view?usp=sharing
Credits:
KaKaRoTo, for ps3Xport
flatz, for the IDPS stealer
Aldo and co., for the PSP2PS3 conversion suite
The dark programmer, for the ps3Xport GUI and Sky Blade Cloud for this guide.
I hope you have enjoyed this and as always you can find me on twitter @riddle43