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Theorem br2base 29454
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 29371 . . . . . . . 8 𝔅 ⊆ 𝒫 ℝ
21sseli 3468 . . . . . . 7 (𝑥 ∈ 𝔅𝑥 ∈ 𝒫 ℝ)
32elpwid 4021 . . . . . 6 (𝑥 ∈ 𝔅𝑥 ⊆ ℝ)
41sseli 3468 . . . . . . 7 (𝑦 ∈ 𝔅𝑦 ∈ 𝒫 ℝ)
54elpwid 4021 . . . . . 6 (𝑦 ∈ 𝔅𝑦 ⊆ ℝ)
6 xpss12 5041 . . . . . 6 ((𝑥 ⊆ ℝ ∧ 𝑦 ⊆ ℝ) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
73, 5, 6syl2an 492 . . . . 5 ((𝑥 ∈ 𝔅𝑦 ∈ 𝔅) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
8 vex 3080 . . . . . . 7 𝑥 ∈ V
9 vex 3080 . . . . . . 7 𝑦 ∈ V
108, 9xpex 6736 . . . . . 6 (𝑥 × 𝑦) ∈ V
1110elpw 4017 . . . . 5 ((𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ) ↔ (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
127, 11sylibr 222 . . . 4 ((𝑥 ∈ 𝔅𝑦 ∈ 𝔅) → (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ))
1312rgen2a 2864 . . 3 𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ)
14 eqid 2514 . . . 4 (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦))
1514rnmpt2ss 28644 . . 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 29372 . . . . . 6 𝔅 = ℝ
18 brsigarn 29370 . . . . . . 7 𝔅 ∈ (sigAlgebra‘ℝ)
19 elrnsiga 29312 . . . . . . 7 (𝔅 ∈ (sigAlgebra‘ℝ) → 𝔅 ran sigAlgebra)
20 unielsiga 29314 . . . . . . 7 (𝔅 ran sigAlgebra → 𝔅 ∈ 𝔅)
2118, 19, 20mp2b 10 . . . . . 6 𝔅 ∈ 𝔅
2217, 21eqeltrri 2589 . . . . 5 ℝ ∈ 𝔅
23 eqid 2514 . . . . 5 (ℝ × ℝ) = (ℝ × ℝ)
24 xpeq1 4946 . . . . . . 7 (𝑥 = ℝ → (𝑥 × 𝑦) = (ℝ × 𝑦))
2524eqeq2d 2524 . . . . . 6 (𝑥 = ℝ → ((ℝ × ℝ) = (𝑥 × 𝑦) ↔ (ℝ × ℝ) = (ℝ × 𝑦)))
26 xpeq2 4947 . . . . . . 7 (𝑦 = ℝ → (ℝ × 𝑦) = (ℝ × ℝ))
2726eqeq2d 2524 . . . . . 6 (𝑦 = ℝ → ((ℝ × ℝ) = (ℝ × 𝑦) ↔ (ℝ × ℝ) = (ℝ × ℝ)))
2825, 27rspc2ev 3199 . . . . 5 ((ℝ ∈ 𝔅 ∧ ℝ ∈ 𝔅 ∧ (ℝ × ℝ) = (ℝ × ℝ)) → ∃𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦))
2922, 22, 23, 28mp3an 1415 . . . 4 𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦)
3014, 10elrnmpt2 6548 . . . 4 ((ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦))
3129, 30mpbir 219 . . 3 (ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦))
32 elpwuni 4447 . . 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 218 1 ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1938  wral 2800  wrex 2801  wss 3444  𝒫 cpw 4011   cuni 4270   × cxp 4930  ran crn 4933  cfv 5689  cmpt2 6428  cr 9690  sigAlgebracsiga 29293  𝔅cbrsiga 29367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-cnex 9747  ax-resscn 9748  ax-pre-lttri 9765  ax-pre-lttrn 9766
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-po 4853  df-so 4854  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-1st 6934  df-2nd 6935  df-er 7505  df-en 7718  df-dom 7719  df-sdom 7720  df-pnf 9831  df-mnf 9832  df-xr 9833  df-ltxr 9834  df-le 9835  df-ioo 11919  df-topgen 15811  df-top 20424  df-bases 20425  df-siga 29294  df-sigagen 29325  df-brsiga 29368
This theorem is referenced by:  sxbrsigalem5  29473  sxbrsiga  29475
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