Theorem br2base 34236

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 34151	. . . . . . . 8 ⊢ 𝔅ℝ ⊆ 𝒫 ℝ
2	1	sseli 3933	. . . . . . 7 ⊢ (𝑥 ∈ 𝔅ℝ → 𝑥 ∈ 𝒫 ℝ)
3	2	elpwid 4562	. . . . . 6 ⊢ (𝑥 ∈ 𝔅ℝ → 𝑥 ⊆ ℝ)
4	1	sseli 3933	. . . . . . 7 ⊢ (𝑦 ∈ 𝔅ℝ → 𝑦 ∈ 𝒫 ℝ)
5	4	elpwid 4562	. . . . . 6 ⊢ (𝑦 ∈ 𝔅ℝ → 𝑦 ⊆ ℝ)
6		xpss12 5638	. . . . . 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 3442	. . . . . . 7 ⊢ 𝑥 ∈ V
9		vex 3442	. . . . . . 7 ⊢ 𝑦 ∈ V
10	8, 9	xpex 7693	. . . . . 6 ⊢ (𝑥 × 𝑦) ∈ V
11	10	elpw 4557	. . . . 5 ⊢ ((𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ) ↔ (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
12	7, 11	sylibr 234	. . . 4 ⊢ ((𝑥 ∈ 𝔅ℝ ∧ 𝑦 ∈ 𝔅ℝ) → (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ))
13	12	rgen2 3169	. . 3 ⊢ ∀𝑥 ∈ 𝔅ℝ ∀𝑦 ∈ 𝔅ℝ (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ)
14		eqid 2729	. . . 4 ⊢ (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦))
15	14	rnmposs 32631	. . 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 34152	. . . . . 6 ⊢ ∪ 𝔅ℝ = ℝ
18		brsigarn 34150	. . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ)
19		elrnsiga 34092	. . . . . . 7 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra)
20		unielsiga 34094	. . . . . . 7 ⊢ (𝔅ℝ ∈ ∪ ran sigAlgebra → ∪ 𝔅ℝ ∈ 𝔅ℝ)
21	18, 19, 20	mp2b 10	. . . . . 6 ⊢ ∪ 𝔅ℝ ∈ 𝔅ℝ
22	17, 21	eqeltrri 2825	. . . . 5 ⊢ ℝ ∈ 𝔅ℝ
23		eqid 2729	. . . . 5 ⊢ (ℝ × ℝ) = (ℝ × ℝ)
24		xpeq1 5637	. . . . . . 7 ⊢ (𝑥 = ℝ → (𝑥 × 𝑦) = (ℝ × 𝑦))
25	24	eqeq2d 2740	. . . . . 6 ⊢ (𝑥 = ℝ → ((ℝ × ℝ) = (𝑥 × 𝑦) ↔ (ℝ × ℝ) = (ℝ × 𝑦)))
26		xpeq2 5644	. . . . . . 7 ⊢ (𝑦 = ℝ → (ℝ × 𝑦) = (ℝ × ℝ))
27	26	eqeq2d 2740	. . . . . 6 ⊢ (𝑦 = ℝ → ((ℝ × ℝ) = (ℝ × 𝑦) ↔ (ℝ × ℝ) = (ℝ × ℝ)))
28	25, 27	rspc2ev 3592	. . . . 5 ⊢ ((ℝ ∈ 𝔅ℝ ∧ ℝ ∈ 𝔅ℝ ∧ (ℝ × ℝ) = (ℝ × ℝ)) → ∃𝑥 ∈ 𝔅ℝ ∃𝑦 ∈ 𝔅ℝ (ℝ × ℝ) = (𝑥 × 𝑦))
29	22, 22, 23, 28	mp3an 1463	. . . 4 ⊢ ∃𝑥 ∈ 𝔅ℝ ∃𝑦 ∈ 𝔅ℝ (ℝ × ℝ) = (𝑥 × 𝑦)
30	14, 10	elrnmpo 7489	. . . 4 ⊢ ((ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ 𝔅ℝ ∃𝑦 ∈ 𝔅ℝ (ℝ × ℝ) = (𝑥 × 𝑦))
31	29, 30	mpbir 231	. . 3 ⊢ (ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦))
32		elpwuni 5057	. . 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 230	1 ⊢ ∪ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3905  𝒫 cpw 4553  ∪ cuni 4861   × cxp 5621  ran crn 5624  ‘cfv 6486   ∈ cmpo 7355  ℝcr 11027  sigAlgebracsiga 34074  𝔅_ℝcbrsiga 34147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-pre-lttri 11102  ax-pre-lttrn 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-po 5531  df-so 5532  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-ioo 13270  df-topgen 17365  df-top 22797  df-bases 22849  df-siga 34075  df-sigagen 34105  df-brsiga 34148

This theorem is referenced by:  sxbrsigalem5  34255  sxbrsiga  34257