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Theorem br2base 30632
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 30549 . . . . . . . 8 𝔅 ⊆ 𝒫 ℝ
21sseli 3732 . . . . . . 7 (𝑥 ∈ 𝔅𝑥 ∈ 𝒫 ℝ)
32elpwid 4306 . . . . . 6 (𝑥 ∈ 𝔅𝑥 ⊆ ℝ)
41sseli 3732 . . . . . . 7 (𝑦 ∈ 𝔅𝑦 ∈ 𝒫 ℝ)
54elpwid 4306 . . . . . 6 (𝑦 ∈ 𝔅𝑦 ⊆ ℝ)
6 xpss12 5273 . . . . . 6 ((𝑥 ⊆ ℝ ∧ 𝑦 ⊆ ℝ) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
73, 5, 6syl2an 495 . . . . 5 ((𝑥 ∈ 𝔅𝑦 ∈ 𝔅) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
8 vex 3335 . . . . . . 7 𝑥 ∈ V
9 vex 3335 . . . . . . 7 𝑦 ∈ V
108, 9xpex 7119 . . . . . 6 (𝑥 × 𝑦) ∈ V
1110elpw 4300 . . . . 5 ((𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ) ↔ (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
127, 11sylibr 224 . . . 4 ((𝑥 ∈ 𝔅𝑦 ∈ 𝔅) → (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ))
1312rgen2a 3107 . . 3 𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ)
14 eqid 2752 . . . 4 (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦))
1514rnmpt2ss 29774 . . 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 30550 . . . . . 6 𝔅 = ℝ
18 brsigarn 30548 . . . . . . 7 𝔅 ∈ (sigAlgebra‘ℝ)
19 elrnsiga 30490 . . . . . . 7 (𝔅 ∈ (sigAlgebra‘ℝ) → 𝔅 ran sigAlgebra)
20 unielsiga 30492 . . . . . . 7 (𝔅 ran sigAlgebra → 𝔅 ∈ 𝔅)
2118, 19, 20mp2b 10 . . . . . 6 𝔅 ∈ 𝔅
2217, 21eqeltrri 2828 . . . . 5 ℝ ∈ 𝔅
23 eqid 2752 . . . . 5 (ℝ × ℝ) = (ℝ × ℝ)
24 xpeq1 5272 . . . . . . 7 (𝑥 = ℝ → (𝑥 × 𝑦) = (ℝ × 𝑦))
2524eqeq2d 2762 . . . . . 6 (𝑥 = ℝ → ((ℝ × ℝ) = (𝑥 × 𝑦) ↔ (ℝ × ℝ) = (ℝ × 𝑦)))
26 xpeq2 5278 . . . . . . 7 (𝑦 = ℝ → (ℝ × 𝑦) = (ℝ × ℝ))
2726eqeq2d 2762 . . . . . 6 (𝑦 = ℝ → ((ℝ × ℝ) = (ℝ × 𝑦) ↔ (ℝ × ℝ) = (ℝ × ℝ)))
2825, 27rspc2ev 3455 . . . . 5 ((ℝ ∈ 𝔅 ∧ ℝ ∈ 𝔅 ∧ (ℝ × ℝ) = (ℝ × ℝ)) → ∃𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦))
2922, 22, 23, 28mp3an 1565 . . . 4 𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦)
3014, 10elrnmpt2 6930 . . . 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 4760 . . 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 1624  wcel 2131  wral 3042  wrex 3043  wss 3707  𝒫 cpw 4294   cuni 4580   × cxp 5256  ran crn 5259  cfv 6041  cmpt2 6807  cr 10119  sigAlgebracsiga 30471  𝔅cbrsiga 30545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-pre-lttri 10194  ax-pre-lttrn 10195
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-po 5179  df-so 5180  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-ioo 12364  df-topgen 16298  df-top 20893  df-bases 20944  df-siga 30472  df-sigagen 30503  df-brsiga 30546
This theorem is referenced by:  sxbrsigalem5  30651  sxbrsiga  30653
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