Theorem imasbas

Description: The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015) (Revised by Mario Carneiro, 11-Jul-2015) (Revised by Thierry Arnoux, 16-Jun-2019) (Revised by AV, 6-Oct-2020)

	Ref	Expression
Hypotheses	imasbas.u	$$⊢\phi \to U=F{“}_{𝑠}R$$
	imasbas.v	$$⊢\phi \to V={\mathrm{Base}}_R$$
	imasbas.f	$$⊢\phi \to F:V\underset{onto}{⟶}B$$
	imasbas.r	$$⊢\phi \to R\in Z$$
Assertion	imasbas	$$⊢\phi \to B={\mathrm{Base}}_U$$

Proof

Step	Hyp	Ref	Expression
1		imasbas.u	$$⊢\phi \to U=F{“}_{𝑠}R$$
2		imasbas.v	$$⊢\phi \to V={\mathrm{Base}}_R$$
3		imasbas.f	$$⊢\phi \to F:V\underset{onto}{⟶}B$$
4		imasbas.r	$$⊢\phi \to R\in Z$$
5		eqid	$$⊢{+}_R={+}_R$$
6		eqid	$$⊢{\cdot }_R={\cdot }_R$$
7		eqid	$$⊢\mathrm{Scalar}\left(R\right)=\mathrm{Scalar}\left(R\right)$$
8		eqid	$$⊢{\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)}={\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)}$$
9		eqid	$$⊢{\cdot }_R={\cdot }_R$$
10		eqid	$$⊢{\cdot }_{𝑖}\left(R\right)={\cdot }_{𝑖}\left(R\right)$$
11		eqid	$$⊢\mathrm{TopOpen}\left(R\right)=\mathrm{TopOpen}\left(R\right)$$
12		eqid	$$⊢\mathrm{dist}\left(R\right)=\mathrm{dist}\left(R\right)$$
13		eqid	$$⊢{\le }_R={\le }_R$$
14		eqidd	$$⊢\phi \to \bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}$$
15		eqidd	$$⊢\phi \to \bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}$$
16		eqidd	$$⊢\phi \to \bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)=\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)$$
17		eqidd	$$⊢\phi \to \bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}=\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}$$
18		eqidd	$$⊢\phi \to \mathrm{TopOpen}\left(R\right)\mathrm{qTop}F=\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F$$
19		eqidd	$$⊢\phi \to \left(x\in B,y\in B⟼sup\left(\bigcup _{n\in \mathbb{N}}\mathrm{ran}\left(g\in \left\{h\in \left({\left(V\times V\right)}^{\left(1\dots n\right)}\right)|\left(F\left(1^{st}\left(h\left(1\right)\right)\right)=x\wedge F\left(2^{nd}\left(h\left(n\right)\right)\right)=y\wedge \forall i\in \left(1\dots n-1\right)F\left(2^{nd}\left(h\left(i\right)\right)\right)=F\left(1^{st}\left(h\left(i+1\right)\right)\right)\right)\right\}⟼{\sum }_{{ℝ}_{𝑠}^{*}}\left(\mathrm{dist}\left(R\right)\circ g\right)\right),{ℝ}^{*},<\right)\right)=\left(x\in B,y\in B⟼sup\left(\bigcup _{n\in \mathbb{N}}\mathrm{ran}\left(g\in \left\{h\in \left({\left(V\times V\right)}^{\left(1\dots n\right)}\right)|\left(F\left(1^{st}\left(h\left(1\right)\right)\right)=x\wedge F\left(2^{nd}\left(h\left(n\right)\right)\right)=y\wedge \forall i\in \left(1\dots n-1\right)F\left(2^{nd}\left(h\left(i\right)\right)\right)=F\left(1^{st}\left(h\left(i+1\right)\right)\right)\right)\right\}⟼{\sum }_{{ℝ}_{𝑠}^{*}}\left(\mathrm{dist}\left(R\right)\circ g\right)\right),{ℝ}^{*},<\right)\right)$$
20		eqidd	$$⊢\phi \to \left(F\circ {\le }_R\right)\circ F^{-1}=\left(F\circ {\le }_R\right)\circ F^{-1}$$
21	1 2 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3 4	imasval	$$⊢\phi \to U=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(x\in B,y\in B⟼sup\left(\bigcup _{n\in \mathbb{N}}\mathrm{ran}\left(g\in \left\{h\in \left({\left(V\times V\right)}^{\left(1\dots n\right)}\right)|\left(F\left(1^{st}\left(h\left(1\right)\right)\right)=x\wedge F\left(2^{nd}\left(h\left(n\right)\right)\right)=y\wedge \forall i\in \left(1\dots n-1\right)F\left(2^{nd}\left(h\left(i\right)\right)\right)=F\left(1^{st}\left(h\left(i+1\right)\right)\right)\right)\right\}⟼{\sum }_{{ℝ}_{𝑠}^{*}}\left(\mathrm{dist}\left(R\right)\circ g\right)\right),{ℝ}^{*},<\right)\right)⟩\right\}$$
22		eqid	$$⊢\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(x\in B,y\in B⟼sup\left(\bigcup _{n\in \mathbb{N}}\mathrm{ran}\left(g\in \left\{h\in \left({\left(V\times V\right)}^{\left(1\dots n\right)}\right)|\left(F\left(1^{st}\left(h\left(1\right)\right)\right)=x\wedge F\left(2^{nd}\left(h\left(n\right)\right)\right)=y\wedge \forall i\in \left(1\dots n-1\right)F\left(2^{nd}\left(h\left(i\right)\right)\right)=F\left(1^{st}\left(h\left(i+1\right)\right)\right)\right)\right\}⟼{\sum }_{{ℝ}_{𝑠}^{*}}\left(\mathrm{dist}\left(R\right)\circ g\right)\right),{ℝ}^{*},<\right)\right)⟩\right\}=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(x\in B,y\in B⟼sup\left(\bigcup _{n\in \mathbb{N}}\mathrm{ran}\left(g\in \left\{h\in \left({\left(V\times V\right)}^{\left(1\dots n\right)}\right)|\left(F\left(1^{st}\left(h\left(1\right)\right)\right)=x\wedge F\left(2^{nd}\left(h\left(n\right)\right)\right)=y\wedge \forall i\in \left(1\dots n-1\right)F\left(2^{nd}\left(h\left(i\right)\right)\right)=F\left(1^{st}\left(h\left(i+1\right)\right)\right)\right)\right\}⟼{\sum }_{{ℝ}_{𝑠}^{*}}\left(\mathrm{dist}\left(R\right)\circ g\right)\right),{ℝ}^{*},<\right)\right)⟩\right\}$$
23	22	imasvalstr	$$⊢\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(x\in B,y\in B⟼sup\left(\bigcup _{n\in \mathbb{N}}\mathrm{ran}\left(g\in \left\{h\in \left({\left(V\times V\right)}^{\left(1\dots n\right)}\right)|\left(F\left(1^{st}\left(h\left(1\right)\right)\right)=x\wedge F\left(2^{nd}\left(h\left(n\right)\right)\right)=y\wedge \forall i\in \left(1\dots n-1\right)F\left(2^{nd}\left(h\left(i\right)\right)\right)=F\left(1^{st}\left(h\left(i+1\right)\right)\right)\right)\right\}⟼{\sum }_{{ℝ}_{𝑠}^{*}}\left(\mathrm{dist}\left(R\right)\circ g\right)\right),{ℝ}^{*},<\right)\right)⟩\right\}\right)\mathrm{Struct}⟨1,12⟩$$
24		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
25		snsstp1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}⟩\right\}$$
26		ssun1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}$$
27	25 26	sstri	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}$$
28		ssun1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(x\in B,y\in B⟼sup\left(\bigcup _{n\in \mathbb{N}}\mathrm{ran}\left(g\in \left\{h\in \left({\left(V\times V\right)}^{\left(1\dots n\right)}\right)|\left(F\left(1^{st}\left(h\left(1\right)\right)\right)=x\wedge F\left(2^{nd}\left(h\left(n\right)\right)\right)=y\wedge \forall i\in \left(1\dots n-1\right)F\left(2^{nd}\left(h\left(i\right)\right)\right)=F\left(1^{st}\left(h\left(i+1\right)\right)\right)\right)\right\}⟼{\sum }_{{ℝ}_{𝑠}^{*}}\left(\mathrm{dist}\left(R\right)\circ g\right)\right),{ℝ}^{*},<\right)\right)⟩\right\}$$
29	27 28	sstri	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{+}_{R}q\right)⟩\right\}⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,F\left(p{\cdot }_{R}q\right)⟩\right\}⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),\mathrm{Scalar}\left(R\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\bigcup _{q\in V}\left(p\in {\mathrm{Base}}_{\mathrm{Scalar}\left(R\right)},x\in \left\{F\left(q\right)\right\}⟼F\left(p{\cdot }_{R}q\right)\right)⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\bigcup _{p\in V}\bigcup _{q\in V}\left\{⟨⟨F\left(p\right),F\left(q\right)⟩,p{\cdot }_{𝑖}\left(R\right)q⟩\right\}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{TopOpen}\left(R\right)\mathrm{qTop}F⟩,⟨{\le }_{\mathrm{ndx}},\left(F\circ {\le }_R\right)\circ F^{-1}⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(x\in B,y\in B⟼sup\left(\bigcup _{n\in \mathbb{N}}\mathrm{ran}\left(g\in \left\{h\in \left({\left(V\times V\right)}^{\left(1\dots n\right)}\right)|\left(F\left(1^{st}\left(h\left(1\right)\right)\right)=x\wedge F\left(2^{nd}\left(h\left(n\right)\right)\right)=y\wedge \forall i\in \left(1\dots n-1\right)F\left(2^{nd}\left(h\left(i\right)\right)\right)=F\left(1^{st}\left(h\left(i+1\right)\right)\right)\right)\right\}⟼{\sum }_{{ℝ}_{𝑠}^{*}}\left(\mathrm{dist}\left(R\right)\circ g\right)\right),{ℝ}^{*},<\right)\right)⟩\right\}$$
30		fvex	$$⊢{\mathrm{Base}}_R\in \mathrm{V}$$
31	2 30	eqeltrdi	$$⊢\phi \to V\in \mathrm{V}$$
32		fornex	$$⊢V\in \mathrm{V}\to \left(F:V\underset{onto}{⟶}B\to B\in \mathrm{V}\right)$$
33	31 3 32	sylc	$$⊢\phi \to B\in \mathrm{V}$$
34		eqid	$$⊢{\mathrm{Base}}_U={\mathrm{Base}}_U$$
35	21 23 24 29 33 34	strfv3	$$⊢\phi \to {\mathrm{Base}}_U=B$$
36	35	eqcomd	$$⊢\phi \to B={\mathrm{Base}}_U$$