Question on understanding a French cartoon

The caption with this 19th century cartoon reads in French:

Quote:

Le Danger De Manger De La Souris Est Qu'Ensuite Votre Chat Ne Coure Après

A internet translation gives me:

'The Danger Of Eating The Mouse Is Then Your Cat Will Run After'

Could a kind bilingual person comment on whether that machine translation is correct?

Thanks

via International Skeptics Forum http://ift.tt/2r0VY10