Theorem br2base 30331

Description: The base set for the generator of the Borel sigma-algebra on ( RR X. RR ) is indeed ( RR X. RR ). (Contributed by Thierry Arnoux, 22-Sep-2017.)

Assertion

Ref	Expression
br2base	|- U. ran ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) = ( RR X. RR )

Distinct variable group:   x, y

Proof of Theorem br2base

Step	Hyp	Ref	Expression
1		brsigasspwrn 30248	. . . . . . . 8 |- 𝔅ℝ C_ ~P RR
2	1	sseli 3599	. . . . . . 7 |- ( x e. 𝔅ℝ -> x e. ~P RR )
3	2	elpwid 4170	. . . . . 6 |- ( x e. 𝔅ℝ -> x C_ RR )
4	1	sseli 3599	. . . . . . 7 |- ( y e. 𝔅ℝ -> y e. ~P RR )
5	4	elpwid 4170	. . . . . 6 |- ( y e. 𝔅ℝ -> y C_ RR )
6		xpss12 5225	. . . . . 6 |- ( ( x C_ RR /\ y C_ RR ) -> ( x X. y ) C_ ( RR X. RR ) )
7	3, 5, 6	syl2an 494	. . . . 5 |- ( ( x e. 𝔅ℝ /\ y e. 𝔅ℝ ) -> ( x X. y ) C_ ( RR X. RR ) )
8		vex 3203	. . . . . . 7 |- x e. _V
9		vex 3203	. . . . . . 7 |- y e. _V
10	8, 9	xpex 6962	. . . . . 6 |- ( x X. y ) e. _V
11	10	elpw 4164	. . . . 5 |- ( ( x X. y ) e. ~P ( RR X. RR ) <-> ( x X. y ) C_ ( RR X. RR ) )
12	7, 11	sylibr 224	. . . 4 |- ( ( x e. 𝔅ℝ /\ y e. 𝔅ℝ ) -> ( x X. y ) e. ~P ( RR X. RR ) )
13	12	rgen2a 2977	. . 3 |- A. x e. 𝔅ℝ A. y e. 𝔅ℝ ( x X. y ) e. ~P ( RR X. RR )
14		eqid 2622	. . . 4 |- ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) = ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) )
15	14	rnmpt2ss 29473	. . 3 |- ( A. x e. 𝔅ℝ A. y e. 𝔅ℝ ( x X. y ) e. ~P ( RR X. RR ) -> ran ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) C_ ~P ( RR X. RR ) )
16	13, 15	ax-mp 5	. 2 |- ran ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) C_ ~P ( RR X. RR )
17		unibrsiga 30249	. . . . . 6 |- U.𝔅ℝ = RR
18		brsigarn 30247	. . . . . . 7 |- 𝔅ℝ e. (sigAlgebra ` RR )
19		elrnsiga 30189	. . . . . . 7 |- (𝔅ℝ e. (sigAlgebra ` RR ) -> 𝔅ℝ e. U. ran sigAlgebra )
20		unielsiga 30191	. . . . . . 7 |- (𝔅ℝ e. U. ran sigAlgebra -> U.𝔅ℝ e. 𝔅ℝ )
21	18, 19, 20	mp2b 10	. . . . . 6 |- U.𝔅ℝ e. 𝔅ℝ
22	17, 21	eqeltrri 2698	. . . . 5 |- RR e. 𝔅ℝ
23		eqid 2622	. . . . 5 |- ( RR X. RR ) = ( RR X. RR )
24		xpeq1 5128	. . . . . . 7 |- ( x = RR -> ( x X. y ) = ( RR X. y ) )
25	24	eqeq2d 2632	. . . . . 6 |- ( x = RR -> ( ( RR X. RR ) = ( x X. y ) <-> ( RR X. RR ) = ( RR X. y ) ) )
26		xpeq2 5129	. . . . . . 7 |- ( y = RR -> ( RR X. y ) = ( RR X. RR ) )
27	26	eqeq2d 2632	. . . . . 6 |- ( y = RR -> ( ( RR X. RR ) = ( RR X. y ) <-> ( RR X. RR ) = ( RR X. RR ) ) )
28	25, 27	rspc2ev 3324	. . . . 5 |- ( ( RR e. 𝔅ℝ /\ RR e. 𝔅ℝ /\ ( RR X. RR ) = ( RR X. RR ) ) -> E. x e. 𝔅ℝ E. y e. 𝔅ℝ ( RR X. RR ) = ( x X. y ) )
29	22, 22, 23, 28	mp3an 1424	. . . 4 |- E. x e. 𝔅ℝ E. y e. 𝔅ℝ ( RR X. RR ) = ( x X. y )
30	14, 10	elrnmpt2 6773	. . . 4 |- ( ( RR X. RR ) e. ran ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) <-> E. x e. 𝔅ℝ E. y e. 𝔅ℝ ( RR X. RR ) = ( x X. y ) )
31	29, 30	mpbir 221	. . 3 |- ( RR X. RR ) e. ran ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) )
32		elpwuni 4616	. . 3 |- ( ( RR X. RR ) e. ran ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) -> ( ran ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) C_ ~P ( RR X. RR ) <-> U. ran ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) = ( RR X. RR ) ) )
33	31, 32	ax-mp 5	. 2 |- ( ran ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) C_ ~P ( RR X. RR ) <-> U. ran ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) = ( RR X. RR ) )
34	16, 33	mpbi 220	1 |- U. ran ( x e. 𝔅ℝ , y e. 𝔅ℝ |-> ( x X. y ) ) = ( RR X. RR )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   X. cxp 5112   ran crn 5115   ` cfv 5888   |-> cmpt2 6652   RRcr 9935  sigAlgebracsiga 30170  𝔅_ℝcbrsiga 30244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioo 12179  df-topgen 16104  df-top 20699  df-bases 20750  df-siga 30171  df-sigagen 30202  df-brsiga 30245

This theorem is referenced by:  sxbrsigalem5  30350  sxbrsiga  30352