The Mekhilta pushes the meat-and-milk prohibition further. What about cooking an animal's flesh in its own milk? Not the mother's milk, not a sister's milk, but the milk the animal itself produced?
The a fortiori argument proceeds in two steps. First, consider the law of shechitah (slaughtering). It is permitted to slaughter two sibling animals on the same day — "fruit" with "fruit" (offspring with offspring) is allowed. But slaughtering an animal and its mother on the same day is forbidden — "fruit" with mother is prohibited.
Now apply this to cooking. Cooking "fruit" in "fruit" — the kid's flesh in another animal's milk — is already established as forbidden. If in the realm of shechitah, where sibling-with-sibling is permitted, parent-with-offspring is still forbidden, then in the realm of cooking, where sibling-with-sibling is already forbidden, parent-with-offspring must be even more thoroughly forbidden.
The final step: if cooking an animal in its mother's milk is forbidden, cooking an animal in its own milk is forbidden too. The animal's relationship to its own milk is even more intimate than the mother's relationship to the kid. Every extension of the prohibition follows logically from the previous one, building an ever-wider circle of forbidden combinations, all anchored in the Torah's original statement about the mother's milk.