Emulation does not work this way.
The PSP and PS1 are using a MIPS CPU which need to be emulated to run on another system. PS1 games are emulated on the PSP and PS Vita, so with a properly made emulator, that would probably run (not even thinking about other problems**)
For the PSP, as it is more powerful than the PS1, can’t say for sure, maybe, I never used a PSP emulator so not sure, but well a PC made emulator will probably not work well. (if not properly optimized)
** like no analog stick, original PS1 game with no need for analog may work with the light key module for the 4 shoulder buttons. PSP games that use the analog stick would be in the exact same situation, and there are way more than the PS1