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Theorem br2base 30459
Description: The base set for the generator of the Borel sigma-algebra on (ℝ × ℝ) is indeed (ℝ × ℝ). (Contributed by Thierry Arnoux, 22-Sep-2017.)
Assertion
Ref Expression
br2base ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)
Distinct variable group:   𝑥,𝑦

Proof of Theorem br2base
StepHypRef Expression
1 brsigasspwrn 30376 . . . . . . . 8 𝔅 ⊆ 𝒫 ℝ
21sseli 3632 . . . . . . 7 (𝑥 ∈ 𝔅𝑥 ∈ 𝒫 ℝ)
32elpwid 4203 . . . . . 6 (𝑥 ∈ 𝔅𝑥 ⊆ ℝ)
41sseli 3632 . . . . . . 7 (𝑦 ∈ 𝔅𝑦 ∈ 𝒫 ℝ)
54elpwid 4203 . . . . . 6 (𝑦 ∈ 𝔅𝑦 ⊆ ℝ)
6 xpss12 5158 . . . . . 6 ((𝑥 ⊆ ℝ ∧ 𝑦 ⊆ ℝ) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
73, 5, 6syl2an 493 . . . . 5 ((𝑥 ∈ 𝔅𝑦 ∈ 𝔅) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
8 vex 3234 . . . . . . 7 𝑥 ∈ V
9 vex 3234 . . . . . . 7 𝑦 ∈ V
108, 9xpex 7004 . . . . . 6 (𝑥 × 𝑦) ∈ V
1110elpw 4197 . . . . 5 ((𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ) ↔ (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
127, 11sylibr 224 . . . 4 ((𝑥 ∈ 𝔅𝑦 ∈ 𝔅) → (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ))
1312rgen2a 3006 . . 3 𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ)
14 eqid 2651 . . . 4 (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦))
1514rnmpt2ss 29601 . . 3 (∀𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ) → ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (ℝ × ℝ))
1613, 15ax-mp 5 . 2 ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (ℝ × ℝ)
17 unibrsiga 30377 . . . . . 6 𝔅 = ℝ
18 brsigarn 30375 . . . . . . 7 𝔅 ∈ (sigAlgebra‘ℝ)
19 elrnsiga 30317 . . . . . . 7 (𝔅 ∈ (sigAlgebra‘ℝ) → 𝔅 ran sigAlgebra)
20 unielsiga 30319 . . . . . . 7 (𝔅 ran sigAlgebra → 𝔅 ∈ 𝔅)
2118, 19, 20mp2b 10 . . . . . 6 𝔅 ∈ 𝔅
2217, 21eqeltrri 2727 . . . . 5 ℝ ∈ 𝔅
23 eqid 2651 . . . . 5 (ℝ × ℝ) = (ℝ × ℝ)
24 xpeq1 5157 . . . . . . 7 (𝑥 = ℝ → (𝑥 × 𝑦) = (ℝ × 𝑦))
2524eqeq2d 2661 . . . . . 6 (𝑥 = ℝ → ((ℝ × ℝ) = (𝑥 × 𝑦) ↔ (ℝ × ℝ) = (ℝ × 𝑦)))
26 xpeq2 5163 . . . . . . 7 (𝑦 = ℝ → (ℝ × 𝑦) = (ℝ × ℝ))
2726eqeq2d 2661 . . . . . 6 (𝑦 = ℝ → ((ℝ × ℝ) = (ℝ × 𝑦) ↔ (ℝ × ℝ) = (ℝ × ℝ)))
2825, 27rspc2ev 3355 . . . . 5 ((ℝ ∈ 𝔅 ∧ ℝ ∈ 𝔅 ∧ (ℝ × ℝ) = (ℝ × ℝ)) → ∃𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦))
2922, 22, 23, 28mp3an 1464 . . . 4 𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦)
3014, 10elrnmpt2 6815 . . . 4 ((ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦))
3129, 30mpbir 221 . . 3 (ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦))
32 elpwuni 4648 . . 3 ((ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) → (ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (ℝ × ℝ) ↔ ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)))
3331, 32ax-mp 5 . 2 (ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (ℝ × ℝ) ↔ ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (ℝ × ℝ))
3416, 33mpbi 220 1 ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wss 3607  𝒫 cpw 4191   cuni 4468   × cxp 5141  ran crn 5144  cfv 5926  cmpt2 6692  cr 9973  sigAlgebracsiga 30298  𝔅cbrsiga 30372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-ioo 12217  df-topgen 16151  df-top 20747  df-bases 20798  df-siga 30299  df-sigagen 30330  df-brsiga 30373
This theorem is referenced by:  sxbrsigalem5  30478  sxbrsiga  30480
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