I want to know how to decide aromaticity of two rings connected by a double bond. For example consider the ring in which when double bond is broken generates tropylium cation and cyclopentadienyl anion. I know that is aromatic because the ions generated are themselves aromatic. How will the aromaticity be affected when we get an antiaromatic one? For example consider the ring in which when we break the double bond cyclopropenyl anion is generated. What will be the overall aromaticity of the molecule in this case. And in general what is the rule to decide the aromaticity of the rings connected by a double bond?
Actually, the breakage of double bond doesn't happen in the way you are thinking it does. Let me illustrate the phenomena what happens in such rings connected by double bonds.
If you have two rings connected by double bonds where the breakage of that will produce anti aromatic rings, the molecule itself becomes very inert towards that breakage. The molecule always tries to gain stability through any perturbations to it. That's why production of anti-aromatic rings becomes extremely difficult in those cases.
In your given case, the molecule becomes highly polar due to unsymmetrical distribution of electron cloud (i.e. more negative charge on cyclopentadienyl ring and more positive charge on the other ring). But in a case, where you are assuming that breakage of double bond gives anti-aromatic cyclopropenyl anion or cyclopentadienyl cation, that breakage becomes extremely impossible. Even if you would subject it to extreme conditions, the molecule will never go into that transition state as that will have immensely high energy. Instead it will always shift towards that transition state by which it can gain some stability, by forming non aromatic or aromatic rings.
If it can't stabilize by that also, it will then become almost inert towards any electrophile or any external reagent which tries to make it anti-aromatic. Even if you perform ozonolysis, it will remain as a perfect keto-group, no unsymmetrical distribution will be observed for these classes of rings.
Considering the number of π-electrons in continuous conjugation of the whole molecule, we can judge if the molecule is aromatic or anti-aromatic. But if the molecules breakage of one double bond gives it anti-aromaticity, it will try to stay in its own form, whatever it may be.