Theorem br2base 34277

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 34193	. . . . . . . 8 ⊢ 𝔅ℝ ⊆ 𝒫 ℝ
2	1	sseli 3930	. . . . . . 7 ⊢ (𝑥 ∈ 𝔅ℝ → 𝑥 ∈ 𝒫 ℝ)
3	2	elpwid 4559	. . . . . 6 ⊢ (𝑥 ∈ 𝔅ℝ → 𝑥 ⊆ ℝ)
4	1	sseli 3930	. . . . . . 7 ⊢ (𝑦 ∈ 𝔅ℝ → 𝑦 ∈ 𝒫 ℝ)
5	4	elpwid 4559	. . . . . 6 ⊢ (𝑦 ∈ 𝔅ℝ → 𝑦 ⊆ ℝ)
6		xpss12 5631	. . . . . 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 3440	. . . . . . 7 ⊢ 𝑥 ∈ V
9		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
10	8, 9	xpex 7686	. . . . . 6 ⊢ (𝑥 × 𝑦) ∈ V
11	10	elpw 4554	. . . . 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 3172	. . 3 ⊢ ∀𝑥 ∈ 𝔅ℝ ∀𝑦 ∈ 𝔅ℝ (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ)
14		eqid 2731	. . . 4 ⊢ (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦))
15	14	rnmposs 32651	. . 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 34194	. . . . . 6 ⊢ ∪ 𝔅ℝ = ℝ
18		brsigarn 34192	. . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ)
19		elrnsiga 34134	. . . . . . 7 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra)
20		unielsiga 34136	. . . . . . 7 ⊢ (𝔅ℝ ∈ ∪ ran sigAlgebra → ∪ 𝔅ℝ ∈ 𝔅ℝ)
21	18, 19, 20	mp2b 10	. . . . . 6 ⊢ ∪ 𝔅ℝ ∈ 𝔅ℝ
22	17, 21	eqeltrri 2828	. . . . 5 ⊢ ℝ ∈ 𝔅ℝ
23		eqid 2731	. . . . 5 ⊢ (ℝ × ℝ) = (ℝ × ℝ)
24		xpeq1 5630	. . . . . . 7 ⊢ (𝑥 = ℝ → (𝑥 × 𝑦) = (ℝ × 𝑦))
25	24	eqeq2d 2742	. . . . . 6 ⊢ (𝑥 = ℝ → ((ℝ × ℝ) = (𝑥 × 𝑦) ↔ (ℝ × ℝ) = (ℝ × 𝑦)))
26		xpeq2 5637	. . . . . . 7 ⊢ (𝑦 = ℝ → (ℝ × 𝑦) = (ℝ × ℝ))
27	26	eqeq2d 2742	. . . . . 6 ⊢ (𝑦 = ℝ → ((ℝ × ℝ) = (ℝ × 𝑦) ↔ (ℝ × ℝ) = (ℝ × ℝ)))
28	25, 27	rspc2ev 3590	. . . . 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 7482	. . . 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 5053	. . 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ⊆ wss 3902  𝒫 cpw 4550  ∪ cuni 4859   × cxp 5614  ran crn 5617  ‘cfv 6481   ∈ cmpo 7348  ℝcr 11002  sigAlgebracsiga 34116  𝔅_ℝcbrsiga 34189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-pre-lttri 11077  ax-pre-lttrn 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-po 5524  df-so 5525  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-ioo 13246  df-topgen 17344  df-top 22807  df-bases 22859  df-siga 34117  df-sigagen 34147  df-brsiga 34190

This theorem is referenced by:  sxbrsigalem5  34296  sxbrsiga  34298