How to prove the existence of an inverse operator?

Question

higher mathematics

Let A, B:E→E, D(A) = D(B) = E. AB + A + I = 0, BA + A + I = 0. Prove that there exists an inverse operator A^−1.
To prove this, we need to show that the image of A is the same as E, and that the kernel of A is zero.
You can show that the kernel is zero, you can do this:
Let Ax \u003d 0, then we substitute this x in the second equation, then we get BAx + Ax + x \u003d 0, that is, x \u003d 0.
But with the proof that the image coincides with E, I didn't really understand how.
That is, we must have the following condition: ∀ y ∈ E, ∃ x ∈ E, Ax = y, then they will coincide.
But here I am a little dumb. If this is substituted by analogy, as above, there is little that can be reasonably obtained.