9 to 10 speed upgrade question

I have a Cervelo Dual with Shimano 9 speed Ultegra that i want to upgrade to match my road wheelsets. What is the minimum parts i need to buy? Here’s my guess:

New bar end shifters to get the 10 speed index for the rear (i do want the index capability)
New chain to match the 10 speed spacing
New front derailleur to match the more narrow chain

Three parts right?
Do i need a new crank?
Do i need a new rear derailleur?

Thanks in advance

You only need to replace the chain and the right bar end shifter. The FD, RD, and cranks will all work.

you shouldn’t need the FD, although the 10sp specific one does run a little nicer AFAIK.

your rear D is fine as is the crank.