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Theorem br2base 30929
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 30846 . . . . . . . 8 𝔅 ⊆ 𝒫 ℝ
21sseli 3817 . . . . . . 7 (𝑥 ∈ 𝔅𝑥 ∈ 𝒫 ℝ)
32elpwid 4391 . . . . . 6 (𝑥 ∈ 𝔅𝑥 ⊆ ℝ)
41sseli 3817 . . . . . . 7 (𝑦 ∈ 𝔅𝑦 ∈ 𝒫 ℝ)
54elpwid 4391 . . . . . 6 (𝑦 ∈ 𝔅𝑦 ⊆ ℝ)
6 xpss12 5370 . . . . . 6 ((𝑥 ⊆ ℝ ∧ 𝑦 ⊆ ℝ) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
73, 5, 6syl2an 589 . . . . 5 ((𝑥 ∈ 𝔅𝑦 ∈ 𝔅) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
8 vex 3401 . . . . . . 7 𝑥 ∈ V
9 vex 3401 . . . . . . 7 𝑦 ∈ V
108, 9xpex 7240 . . . . . 6 (𝑥 × 𝑦) ∈ V
1110elpw 4385 . . . . 5 ((𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ) ↔ (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
127, 11sylibr 226 . . . 4 ((𝑥 ∈ 𝔅𝑦 ∈ 𝔅) → (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ))
1312rgen2a 3159 . . 3 𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ)
14 eqid 2778 . . . 4 (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦))
1514rnmpt2ss 30039 . . 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 30847 . . . . . 6 𝔅 = ℝ
18 brsigarn 30845 . . . . . . 7 𝔅 ∈ (sigAlgebra‘ℝ)
19 elrnsiga 30787 . . . . . . 7 (𝔅 ∈ (sigAlgebra‘ℝ) → 𝔅 ran sigAlgebra)
20 unielsiga 30789 . . . . . . 7 (𝔅 ran sigAlgebra → 𝔅 ∈ 𝔅)
2118, 19, 20mp2b 10 . . . . . 6 𝔅 ∈ 𝔅
2217, 21eqeltrri 2856 . . . . 5 ℝ ∈ 𝔅
23 eqid 2778 . . . . 5 (ℝ × ℝ) = (ℝ × ℝ)
24 xpeq1 5369 . . . . . . 7 (𝑥 = ℝ → (𝑥 × 𝑦) = (ℝ × 𝑦))
2524eqeq2d 2788 . . . . . 6 (𝑥 = ℝ → ((ℝ × ℝ) = (𝑥 × 𝑦) ↔ (ℝ × ℝ) = (ℝ × 𝑦)))
26 xpeq2 5376 . . . . . . 7 (𝑦 = ℝ → (ℝ × 𝑦) = (ℝ × ℝ))
2726eqeq2d 2788 . . . . . 6 (𝑦 = ℝ → ((ℝ × ℝ) = (ℝ × 𝑦) ↔ (ℝ × ℝ) = (ℝ × ℝ)))
2825, 27rspc2ev 3526 . . . . 5 ((ℝ ∈ 𝔅 ∧ ℝ ∈ 𝔅 ∧ (ℝ × ℝ) = (ℝ × ℝ)) → ∃𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦))
2922, 22, 23, 28mp3an 1534 . . . 4 𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦)
3014, 10elrnmpt2 7050 . . . 4 ((ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ 𝔅𝑦 ∈ 𝔅 (ℝ × ℝ) = (𝑥 × 𝑦))
3129, 30mpbir 223 . . 3 (ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦))
32 elpwuni 4850 . . 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 222 1 ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1601  wcel 2107  wral 3090  wrex 3091  wss 3792  𝒫 cpw 4379   cuni 4671   × cxp 5353  ran crn 5356  cfv 6135  cmpt2 6924  cr 10271  sigAlgebracsiga 30768  𝔅cbrsiga 30842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-pre-lttri 10346  ax-pre-lttrn 10347
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-po 5274  df-so 5275  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-ioo 12491  df-topgen 16490  df-top 21106  df-bases 21158  df-siga 30769  df-sigagen 30800  df-brsiga 30843
This theorem is referenced by:  sxbrsigalem5  30948  sxbrsiga  30950
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