The tautological degree-one class is well defined and fiber generating

Let $c,c^{\mathrm{\prime }}:P(E)\to {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ be any two classifying maps as in the definition. They have isomorphic tautological pullbacks, both isomorphic to ${\gamma }_E$. By the projective definition $P(E)$ is paracompact Hausdorff CGWH and ${\gamma }_E$ is numerable, so injectivity of the bijection [F1] gives equality of the actual unbased homotopy classes $[c]=[c^{\mathrm{\prime }}]$. Thus [F2] gives $c^{\ast }a=c^{\mathrm{\prime }\ast }a$. This proves independence for all classifying maps, without limiting them to any particular embedding construction. AC is inherited from the stated projective and classification interfaces.

Restriction to a fiber. Fix $b\in B$ and identify the fiber $P(E_b)$ with ${\mathbb{R}\mathbb{P}}^{n-1}$ through a linear isomorphism $E_b\cong {\mathbb{R}}^n$; write $i:P(E_b)\hookrightarrow P(E)$ for the inclusion. By [F7] the pullback $i^{\ast }{\gamma }_E$ is the tautological line ${\gamma }^1$ over $P(E_b)\cong {\mathbb{R}\mathbb{P}}^{n-1}$, and the standard inclusion $j:{\mathbb{R}\mathbb{P}}^{n-1}\hookrightarrow {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ satisfies $j^{\ast }{\gamma }_1\cong {\gamma }^1$ by [F5]. On the other hand $(ci{)}^{\ast }{\gamma }_1\cong i^{\ast }{c}^{\ast }{\gamma }_1\cong i^{\ast }{\gamma }_E\cong {\gamma }^1$, so $ci$ and $j$ are two maps to ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ with isomorphic pullbacks of the tautological line. The fiber is a compact Hausdorff finite CW complex, hence paracompact by [F8] and CGWH. Its tautological line is numerable by the same finite coordinate partition used in the projective definition. Injectivity of the classification bijection [F1] therefore gives an actual homotopy $ci\simeq j$ on this fiber. Therefore [F2] and the definition of $x_E$ give $i^{\ast }{x}_E=(ci{)}^{\ast }a=j^{\ast }a,$ which is the standard generator of $H^1({\mathbb{R}\mathbb{P}}^{n-1};{\mathbb{F}}_2)$ when $n-1\ge 1$ and zero when $n-1=0$, by [F3]. This proves clause 2.

Fiber basis. Restriction $i^{\ast }:H^{\ast }(P(E);{\mathbb{F}}_2)\to H^{\ast }(P(E_b);{\mathbb{F}}_2)$ is a unital ring homomorphism, so [F6] gives $i^{\ast }(x_E^k)=(i^{\ast }{x}_E{)}^k$. By step 2.1 this is $a^k$ for every $k\ge 0$, with the convention that $a^0=1$ and that $a=0$ when $n=1$; in particular $i^{\ast }(1)=1$. By [F3], restriction is an isomorphism in every degree from zero to $n-1$, so its images $1,a,\dots ,a^{n-1}$ are nonzero and span their respective one-dimensional groups. By [F4] there are no groups in higher degrees. They therefore form an ${\mathbb{F}}_2$-basis; they are the restrictions of $1,x_E,\dots ,x_E^{n-1}$. For $n=1$ the fiber is ${\mathbb{R}\mathbb{P}}^0$, a point, and the list reduces to $1$. For $n\ge 2$ the class $i^{\ast }{x}_E=a$ is nonzero and generates $H^1$ of the fiber. This proves clause 3.