Theorem br2base 32236

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 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)

Distinct variable group:   𝑥,𝑦

Proof of Theorem br2base

Step	Hyp	Ref	Expression
1		brsigasspwrn 32153	. . . . . . . 8 ⊢ 𝔅ℝ ⊆ 𝒫 ℝ
2	1	sseli 3917	. . . . . . 7 ⊢ (𝑥 ∈ 𝔅ℝ → 𝑥 ∈ 𝒫 ℝ)
3	2	elpwid 4544	. . . . . 6 ⊢ (𝑥 ∈ 𝔅ℝ → 𝑥 ⊆ ℝ)
4	1	sseli 3917	. . . . . . 7 ⊢ (𝑦 ∈ 𝔅ℝ → 𝑦 ∈ 𝒫 ℝ)
5	4	elpwid 4544	. . . . . 6 ⊢ (𝑦 ∈ 𝔅ℝ → 𝑦 ⊆ ℝ)
6		xpss12 5604	. . . . . 6 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑦 ⊆ ℝ) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
7	3, 5, 6	syl2an 596	. . . . 5 ⊢ ((𝑥 ∈ 𝔅ℝ ∧ 𝑦 ∈ 𝔅ℝ) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
8		vex 3436	. . . . . . 7 ⊢ 𝑥 ∈ V
9		vex 3436	. . . . . . 7 ⊢ 𝑦 ∈ V
10	8, 9	xpex 7603	. . . . . 6 ⊢ (𝑥 × 𝑦) ∈ V
11	10	elpw 4537	. . . . 5 ⊢ ((𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ) ↔ (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
12	7, 11	sylibr 233	. . . 4 ⊢ ((𝑥 ∈ 𝔅ℝ ∧ 𝑦 ∈ 𝔅ℝ) → (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ))
13	12	rgen2 3120	. . 3 ⊢ ∀𝑥 ∈ 𝔅ℝ ∀𝑦 ∈ 𝔅ℝ (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ)
14		eqid 2738	. . . 4 ⊢ (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦))
15	14	rnmposs 31011	. . 3 ⊢ (∀𝑥 ∈ 𝔅ℝ ∀𝑦 ∈ 𝔅ℝ (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ) → ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (ℝ × ℝ))
16	13, 15	ax-mp 5	. 2 ⊢ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (ℝ × ℝ)
17		unibrsiga 32154	. . . . . 6 ⊢ ∪ 𝔅ℝ = ℝ
18		brsigarn 32152	. . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ)
19		elrnsiga 32094	. . . . . . 7 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra)
20		unielsiga 32096	. . . . . . 7 ⊢ (𝔅ℝ ∈ ∪ ran sigAlgebra → ∪ 𝔅ℝ ∈ 𝔅ℝ)
21	18, 19, 20	mp2b 10	. . . . . 6 ⊢ ∪ 𝔅ℝ ∈ 𝔅ℝ
22	17, 21	eqeltrri 2836	. . . . 5 ⊢ ℝ ∈ 𝔅ℝ
23		eqid 2738	. . . . 5 ⊢ (ℝ × ℝ) = (ℝ × ℝ)
24		xpeq1 5603	. . . . . . 7 ⊢ (𝑥 = ℝ → (𝑥 × 𝑦) = (ℝ × 𝑦))
25	24	eqeq2d 2749	. . . . . 6 ⊢ (𝑥 = ℝ → ((ℝ × ℝ) = (𝑥 × 𝑦) ↔ (ℝ × ℝ) = (ℝ × 𝑦)))
26		xpeq2 5610	. . . . . . 7 ⊢ (𝑦 = ℝ → (ℝ × 𝑦) = (ℝ × ℝ))
27	26	eqeq2d 2749	. . . . . 6 ⊢ (𝑦 = ℝ → ((ℝ × ℝ) = (ℝ × 𝑦) ↔ (ℝ × ℝ) = (ℝ × ℝ)))
28	25, 27	rspc2ev 3572	. . . . 5 ⊢ ((ℝ ∈ 𝔅ℝ ∧ ℝ ∈ 𝔅ℝ ∧ (ℝ × ℝ) = (ℝ × ℝ)) → ∃𝑥 ∈ 𝔅ℝ ∃𝑦 ∈ 𝔅ℝ (ℝ × ℝ) = (𝑥 × 𝑦))
29	22, 22, 23, 28	mp3an 1460	. . . 4 ⊢ ∃𝑥 ∈ 𝔅ℝ ∃𝑦 ∈ 𝔅ℝ (ℝ × ℝ) = (𝑥 × 𝑦)
30	14, 10	elrnmpo 7410	. . . 4 ⊢ ((ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ 𝔅ℝ ∃𝑦 ∈ 𝔅ℝ (ℝ × ℝ) = (𝑥 × 𝑦))
31	29, 30	mpbir 230	. . 3 ⊢ (ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦))
32		elpwuni 5034	. . 3 ⊢ ((ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) → (ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (ℝ × ℝ) ↔ ∪ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)))
33	31, 32	ax-mp 5	. 2 ⊢ (ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (ℝ × ℝ) ↔ ∪ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (ℝ × ℝ))
34	16, 33	mpbi 229	1 ⊢ ∪ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839   × cxp 5587  ran crn 5590  ‘cfv 6433   ∈ cmpo 7277  ℝcr 10870  sigAlgebracsiga 32076  𝔅_ℝcbrsiga 32149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-pre-lttri 10945  ax-pre-lttrn 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-ioo 13083  df-topgen 17154  df-top 22043  df-bases 22096  df-siga 32077  df-sigagen 32107  df-brsiga 32150

This theorem is referenced by:  sxbrsigalem5  32255  sxbrsiga  32257