Let $f:\widetilde X\rightarrow Y$ be a proper morphism between smooth complex varieties which is birational unto its image $X=f(X)$. Assume the singular locus $W\subset X$ of $X$ is smooth and that $f_{f^{-1}(W)}:f^{-1}(W)\rightarrow W$ is smooth with connected fibers.
Do we have an inclusion (at least set-theoretic) between the exceptional divisors $\mathbb P(N_{f^{-1}(W)/\widetilde X})\subset \mathbb P(N_{W/Y})$ ?
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