Theorem br2base 34301

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 34216	. . . . . . . 8 ⊢ 𝔅ℝ ⊆ 𝒫 ℝ
2	1	sseli 3954	. . . . . . 7 ⊢ (𝑥 ∈ 𝔅ℝ → 𝑥 ∈ 𝒫 ℝ)
3	2	elpwid 4584	. . . . . 6 ⊢ (𝑥 ∈ 𝔅ℝ → 𝑥 ⊆ ℝ)
4	1	sseli 3954	. . . . . . 7 ⊢ (𝑦 ∈ 𝔅ℝ → 𝑦 ∈ 𝒫 ℝ)
5	4	elpwid 4584	. . . . . 6 ⊢ (𝑦 ∈ 𝔅ℝ → 𝑦 ⊆ ℝ)
6		xpss12 5669	. . . . . 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 3463	. . . . . . 7 ⊢ 𝑥 ∈ V
9		vex 3463	. . . . . . 7 ⊢ 𝑦 ∈ V
10	8, 9	xpex 7747	. . . . . 6 ⊢ (𝑥 × 𝑦) ∈ V
11	10	elpw 4579	. . . . 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 3184	. . 3 ⊢ ∀𝑥 ∈ 𝔅ℝ ∀𝑦 ∈ 𝔅ℝ (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ)
14		eqid 2735	. . . 4 ⊢ (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦))
15	14	rnmposs 32652	. . 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 34217	. . . . . 6 ⊢ ∪ 𝔅ℝ = ℝ
18		brsigarn 34215	. . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ)
19		elrnsiga 34157	. . . . . . 7 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra)
20		unielsiga 34159	. . . . . . 7 ⊢ (𝔅ℝ ∈ ∪ ran sigAlgebra → ∪ 𝔅ℝ ∈ 𝔅ℝ)
21	18, 19, 20	mp2b 10	. . . . . 6 ⊢ ∪ 𝔅ℝ ∈ 𝔅ℝ
22	17, 21	eqeltrri 2831	. . . . 5 ⊢ ℝ ∈ 𝔅ℝ
23		eqid 2735	. . . . 5 ⊢ (ℝ × ℝ) = (ℝ × ℝ)
24		xpeq1 5668	. . . . . . 7 ⊢ (𝑥 = ℝ → (𝑥 × 𝑦) = (ℝ × 𝑦))
25	24	eqeq2d 2746	. . . . . 6 ⊢ (𝑥 = ℝ → ((ℝ × ℝ) = (𝑥 × 𝑦) ↔ (ℝ × ℝ) = (ℝ × 𝑦)))
26		xpeq2 5675	. . . . . . 7 ⊢ (𝑦 = ℝ → (ℝ × 𝑦) = (ℝ × ℝ))
27	26	eqeq2d 2746	. . . . . 6 ⊢ (𝑦 = ℝ → ((ℝ × ℝ) = (ℝ × 𝑦) ↔ (ℝ × ℝ) = (ℝ × ℝ)))
28	25, 27	rspc2ev 3614	. . . . 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 7543	. . . 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 5081	. . 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 2108  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  𝒫 cpw 4575  ∪ cuni 4883   × cxp 5652  ran crn 5655  ‘cfv 6531   ∈ cmpo 7407  ℝcr 11128  sigAlgebracsiga 34139  𝔅_ℝcbrsiga 34212

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-pre-lttri 11203  ax-pre-lttrn 11204

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-ioo 13366  df-topgen 17457  df-top 22832  df-bases 22884  df-siga 34140  df-sigagen 34170  df-brsiga 34213

This theorem is referenced by:  sxbrsigalem5  34320  sxbrsiga  34322