Can Mockito verify an argument has certain properties/fields?

In Mockito 2.1.0 and up with Java 8 you can pass the lambda to argThat out of the box so that one does not need a custom argument matchers. For the example in the OP would be:

verify(mockedFoo).doThing(argThat((Bar aBar) -> aBar.getI() == 5));

This is because as of Mockito 2.1.0, ArgumentMatcher is a functional interface.