Assume that operators $A,B$ and $C$ are Fredholm where operator $A:= (B, C)$ mapping from vector spaces $X \rightarrow Y \oplus Z$, $B: X \rightarrow Y$ and $C: X \rightarrow Z$. Now we restrict operator $B$ to a smaller domain $$B_{1}: ker(C) \rightarrow Y.$$
Is it true or not that $dim((Y \oplus Z) / Img(A)) = dim(Y / Img(B_1))$?