No that's partially incorrect.The purpose of dilithium in warp engines is to contain antimatter. Antimatter is the fuel.
Dilithium doesn't "contain" the Antimatter, it is used to focus the flow of the the Plasma Stream within the warp core to vastly improve how the warp engines work.
That's why Dilithium has to be "reconstituted" after awhile, the plasma degrades the dilithium crystals.
The Antimatter is collected separately and held in "Antimatter Containment Vessels" which are connected to the system and are combined with a Matter stream to create the combined "Plasma Stream" which then passes through the dilithium crystals before going to the Warp Engines to "energize" the WARP Coils.
Dilithium is actually needed to achieve the higher warp speeds, beyond warp 2.
That's also why the Phoenix didn't need it, Cochranes initial Warp engine was designed to only be able to reach about Warp 1.5.
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