The blowup is an isomorphism off the center

Cover $X$ by affine opens $U=\mathrm{Spec}A$ with $\mathcal{I}{\mathrm{\mid }}_U=(f_0,\dots ,f_r)$; by [F1] and [F3] the preimage ${\pi }^{-1}(U)$ is covered by the charts $\mathrm{Spec}A[I\mathrm{/}{f}_i]$, and by [F2] the chart ring localizes to $A_{f_i}$ after inverting $f_i$, and this chart contains the entire inverse image of $D(f_i)$. Indeed, in every chart $j$ one has $f_i=f_j(f_i\mathrm{/}{f}_j)$; wherever $f_i$ is invertible, both $f_j$ and the ratio $f_i\mathrm{/}{f}_j$ are invertible. The ratio-overlap formula of [F1] puts this open of chart $j$ in chart $i$. Thus the restriction of $\pi$ over the principal open $D(f_i)$ is an isomorphism $D(f_i)\to D(f_i)$: it is the structural map $\mathrm{Spec}A[I\mathrm{/}{f}_i]\to \mathrm{Spec}A$ followed by localization, and $(A[I\mathrm{/}{f}_i]{)}_{f_i}=A_{f_i}$.

These local inverses glue. Let $O$ and $O\mathrm{\prime }$ be any two base opens of the form $D(f_i)$ in step 1.1, possibly in different affine base neighborhoods. The structural map on the whole inverse image ${\pi }^{-1}(O)$ is an isomorphism onto $O$. Restricting it to $O\cap O\mathrm{\prime }$ gives an isomorphism ${\pi }^{-1}(O\cap O\mathrm{\prime })\to O\cap O\mathrm{\prime }$. Both local inverse maps restricted to this intersection land in that inverse image and are inverses of this same isomorphism, so they agree. Thus they glue to $\sigma :X\setminus Z\to W:={\pi }^{-1}(X\setminus Z)$ with $\pi \circ \sigma =\mathrm{id}$. For two charts in one affine base, only the restriction of their chart overlap over $D(f_i{f}_j)$ is identified with $D(f_i{f}_j)$; the whole ratio overlap can also contain points over $Z$.

The morphism $\sigma$ is an inverse for $\pi {\mathrm{\mid }}_W$. Since $X\setminus Z$ is covered by the opens $D(f_i)$ of step 1.1, and over each such open the restriction of $\sigma$ is the inverse of the restriction of $\pi$ (step 1.1), the composite $\sigma \circ \pi {\mathrm{\mid }}_W$ agrees with ${\mathrm{id}}_W$ after restriction to the cover of $W$ by the opens ${\pi }^{-1}(D(f_i))\cap \mathrm{Spec}A[I\mathrm{/}{f}_i]$, on each of which $\pi$ is an isomorphism; hence $\sigma \circ \pi {\mathrm{\mid }}_W={\mathrm{id}}_W$ and $\pi {\mathrm{\mid }}_W\circ \sigma ={\mathrm{id}}_{X\setminus Z}$, so $\pi {\mathrm{\mid }}_W$ is an isomorphism.

The inverse is characterized by the universal property: the inverse image of $Z$ under the identity $X\setminus Z\to X$ is empty, hence the zero Cartier divisor, which is effective by [F5]; so [F4] gives a unique $X$-morphism $\tau \text{\hspace{0.17em}⁣}:X\setminus Z\to {\mathrm{Bl}}_{\mathcal{I}}X$ lifting the identity, and $\tau$ is an inverse of $\pi$ over $X\setminus Z$; by uniqueness of the inverse of the isomorphism $\pi {\mathrm{\mid }}_W$ of step 3.1, $\tau =\sigma$. In particular $\sigma$ is the unique morphism over $X$ from $X\setminus Z$ into the blowup.

Finally $E$ is the complement of $W$: by [F6], $E={\pi }^{-1}(Z)$ is set-theoretically the preimage of $Z$, so its underlying set is the complement of the underlying set of ${\pi }^{-1}(X\setminus Z)=W$, i.e. $E$ is the complement of the open subscheme $W$ in the blowup.