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Theorem br2base 34251
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 34166 . . . . . . . 8 𝔅 ⊆ 𝒫 ℝ
21sseli 3991 . . . . . . 7 (𝑥 ∈ 𝔅𝑥 ∈ 𝒫 ℝ)
32elpwid 4614 . . . . . 6 (𝑥 ∈ 𝔅𝑥 ⊆ ℝ)
41sseli 3991 . . . . . . 7 (𝑦 ∈ 𝔅𝑦 ∈ 𝒫 ℝ)
54elpwid 4614 . . . . . 6 (𝑦 ∈ 𝔅𝑦 ⊆ ℝ)
6 xpss12 5704 . . . . . 6 ((𝑥 ⊆ ℝ ∧ 𝑦 ⊆ ℝ) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
73, 5, 6syl2an 596 . . . . 5 ((𝑥 ∈ 𝔅𝑦 ∈ 𝔅) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
8 vex 3482 . . . . . . 7 𝑥 ∈ V
9 vex 3482 . . . . . . 7 𝑦 ∈ V
108, 9xpex 7772 . . . . . 6 (𝑥 × 𝑦) ∈ V
1110elpw 4609 . . . . 5 ((𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ) ↔ (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
127, 11sylibr 234 . . . 4 ((𝑥 ∈ 𝔅𝑦 ∈ 𝔅) → (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ))
1312rgen2 3197 . . 3 𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ)
14 eqid 2735 . . . 4 (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦))
1514rnmposs 32691 . . 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 34167 . . . . . 6 𝔅 = ℝ
18 brsigarn 34165 . . . . . . 7 𝔅 ∈ (sigAlgebra‘ℝ)
19 elrnsiga 34107 . . . . . . 7 (𝔅 ∈ (sigAlgebra‘ℝ) → 𝔅 ran sigAlgebra)
20 unielsiga 34109 . . . . . . 7 (𝔅 ran sigAlgebra → 𝔅 ∈ 𝔅)
2118, 19, 20mp2b 10 . . . . . 6 𝔅 ∈ 𝔅
2217, 21eqeltrri 2836 . . . . 5 ℝ ∈ 𝔅
23 eqid 2735 . . . . 5 (ℝ × ℝ) = (ℝ × ℝ)
24 xpeq1 5703 . . . . . . 7 (𝑥 = ℝ → (𝑥 × 𝑦) = (ℝ × 𝑦))
2524eqeq2d 2746 . . . . . 6 (𝑥 = ℝ → ((ℝ × ℝ) = (𝑥 × 𝑦) ↔ (ℝ × ℝ) = (ℝ × 𝑦)))
26 xpeq2 5710 . . . . . . 7 (𝑦 = ℝ → (ℝ × 𝑦) = (ℝ × ℝ))
2726eqeq2d 2746 . . . . . 6 (𝑦 = ℝ → ((ℝ × ℝ) = (ℝ × 𝑦) ↔ (ℝ × ℝ) = (ℝ × ℝ)))
2825, 27rspc2ev 3635 . . . . 5 ((ℝ ∈ 𝔅 ∧ ℝ ∈ 𝔅 ∧ (ℝ × ℝ) = (ℝ × ℝ)) → ∃𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦))
2922, 22, 23, 28mp3an 1460 . . . 4 𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦)
3014, 10elrnmpo 7569 . . . 4 ((ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦))
3129, 30mpbir 231 . . 3 (ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦))
32 elpwuni 5110 . . 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 230 1 ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  wss 3963  𝒫 cpw 4605   cuni 4912   × cxp 5687  ran crn 5690  cfv 6563  cmpo 7433  cr 11152  sigAlgebracsiga 34089  𝔅cbrsiga 34162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-ioo 13388  df-topgen 17490  df-top 22916  df-bases 22969  df-siga 34090  df-sigagen 34120  df-brsiga 34163
This theorem is referenced by:  sxbrsigalem5  34270  sxbrsiga  34272
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