I am currently attempting to install a reverse osmosis water filtration system in the cabinet underneath my kitchen sink next to a refrigerator. The system comes with a faucet which appears to get it's water supply from the blue 1/4 inch plastic tubing. I am not too fond of the idea of having a second water faucet on our kitchen sink and would much rather have the reverse osmosis water system feed the refrigerator so that our ice and and water from the fridge are filtered.
I looked underneath my kitchen sink and found that on the main cold water pipe, there is what appears to be a self piercing T-tap and that one of the lines goes through the cabinet and likely behind a dishwasher. I do not see any other water lines leading there besides the hot water line. I'm assuming that the cold water line feeds the dishwasher and there might be another T-tap splitting the water going to the dishwasher and another line that will feed the 1/4 inch water supply line that goes to my refrigerator.
I was wondering would it be possible to just run a much longer blue 1/4 inch tube from the RO system to my refrigerator? (G to H in the below diagram containing the filtered water). I think I would rather just have the filtered water go straight to the refrigerator.
UPDATE: I pulled my refrigerator out and found that the tap on the cold water pipe goes directly to the refrigerator. What would be the best way to connect a 1/4 inch plastic tubing from the filtered water directly to the refrigerator? Is there a way for me to just directly shut off the T-tap on the cold water valve and just wire the 1/4 inch line directly from the end of the water filtration system directly to my refrigerator?
EDIT #2 – I think what would solve my issue is if I could somehow cut the water supply to that line and then use that angle stop valve to feed the RO filter and then use the blue line at the end of my RO filter to be wired directly into my refrigerator.
Is there a way for me to cut the water to braided line with a red line through it and then have the water from the saddle valve travel exclusively through the 1/4 inch plastic line? Or is there a better solution than that? 
Also, could I just take out that blue 1/4 inch plastic line from the top of the refrigerator even though it is powered by the steel braided hose on the bottom of the refrigerator?
(rear of refrigerator) 
Cold water tap for refrigerator water dispenser and possibly dishwasher
Reverse Osmosis diagram
Top of dishwasher
Best Answer
Add this to the top og the saddle valve, it will allow you to add the intake line for the RO, then route you RO line to the icemaker.