Set in a set notation

I'm not sure how to clearly express this informally, but

$A$ is a set containing cars, and $B$ is a set containing parts. But it is also so that $A_1$'s $B$ set is not the same set as $A_2$'s $B$ set, etc.

I could also use the following example to explain what I mean; $A$ is set of moms and $B$ are sets of children that a specific mom in the $A$ set has. Lets say $A_1$ is 'Kari' and $A_2$ is 'Lisa', 'Kari' is the mother of 'Karl' & 'Tia', while 'Lisa' is the mother of 'Bill'. The $B$ set does not contain all the children regardless of mother, but each mother has her own $B$ set.

How can I state using something like this using Set Notation ?

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