Theorem br2base 34884

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 34800	. . . . . . . 8 ⊢ 𝔅ℝ ⊆ 𝒫 ℝ
2	1	sseli 3927	. . . . . . 7 ⊢ (𝑥 ∈ 𝔅ℝ → 𝑥 ∈ 𝒫 ℝ)
3	2	elpwid 4566	. . . . . 6 ⊢ (𝑥 ∈ 𝔅ℝ → 𝑥 ⊆ ℝ)
4	1	sseli 3927	. . . . . . 7 ⊢ (𝑦 ∈ 𝔅ℝ → 𝑦 ∈ 𝒫 ℝ)
5	4	elpwid 4566	. . . . . 6 ⊢ (𝑦 ∈ 𝔅ℝ → 𝑦 ⊆ ℝ)
6		xpss12 5666	. . . . . 6 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑦 ⊆ ℝ) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
7	3, 5, 6	syl2an 608	. . . . 5 ⊢ ((𝑥 ∈ 𝔅ℝ ∧ 𝑦 ∈ 𝔅ℝ) → (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
8		vex 3455	. . . . . . 7 ⊢ 𝑥 ∈ V
9		vex 3455	. . . . . . 7 ⊢ 𝑦 ∈ V
10	8, 9	xpex 7756	. . . . . 6 ⊢ (𝑥 × 𝑦) ∈ V
11	10	elpw 4561	. . . . 5 ⊢ ((𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ) ↔ (𝑥 × 𝑦) ⊆ (ℝ × ℝ))
12	7, 11	sylibr 237	. . . 4 ⊢ ((𝑥 ∈ 𝔅ℝ ∧ 𝑦 ∈ 𝔅ℝ) → (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ))
13	12	rgen2 3203	. . 3 ⊢ ∀𝑥 ∈ 𝔅ℝ ∀𝑦 ∈ 𝔅ℝ (𝑥 × 𝑦) ∈ 𝒫 (ℝ × ℝ)
14		eqid 2761	. . . 4 ⊢ (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦))
15	14	rnmposs 33249	. . 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 34801	. . . . . 6 ⊢ ∪ 𝔅ℝ = ℝ
18		brsigarn 34799	. . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ)
19		elrnsiga 34740	. . . . . . 7 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra)
20		unielsiga 34742	. . . . . . 7 ⊢ (𝔅ℝ ∈ ∪ ran sigAlgebra → ∪ 𝔅ℝ ∈ 𝔅ℝ)
21	18, 19, 20	mp2b 10	. . . . . 6 ⊢ ∪ 𝔅ℝ ∈ 𝔅ℝ
22	17, 21	eqeltrri 2858	. . . . 5 ⊢ ℝ ∈ 𝔅ℝ
23		eqid 2761	. . . . 5 ⊢ (ℝ × ℝ) = (ℝ × ℝ)
24		xpeq1 5665	. . . . . . 7 ⊢ (𝑥 = ℝ → (𝑥 × 𝑦) = (ℝ × 𝑦))
25	24	eqeq2d 2772	. . . . . 6 ⊢ (𝑥 = ℝ → ((ℝ × ℝ) = (𝑥 × 𝑦) ↔ (ℝ × ℝ) = (ℝ × 𝑦)))
26		xpeq2 5672	. . . . . . 7 ⊢ (𝑦 = ℝ → (ℝ × 𝑦) = (ℝ × ℝ))
27	26	eqeq2d 2772	. . . . . 6 ⊢ (𝑦 = ℝ → ((ℝ × ℝ) = (ℝ × 𝑦) ↔ (ℝ × ℝ) = (ℝ × ℝ)))
28	25, 27	rspc2ev 3589	. . . . 5 ⊢ ((ℝ ∈ 𝔅ℝ ∧ ℝ ∈ 𝔅ℝ ∧ (ℝ × ℝ) = (ℝ × ℝ)) → ∃𝑥 ∈ 𝔅ℝ ∃𝑦 ∈ 𝔅ℝ (ℝ × ℝ) = (𝑥 × 𝑦))
29	22, 22, 23, 28	mp3an 1490	. . . 4 ⊢ ∃𝑥 ∈ 𝔅ℝ ∃𝑦 ∈ 𝔅ℝ (ℝ × ℝ) = (𝑥 × 𝑦)
30	14, 10	elrnmpo 7548	. . . 4 ⊢ ((ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ 𝔅ℝ ∃𝑦 ∈ 𝔅ℝ (ℝ × ℝ) = (𝑥 × 𝑦))
31	29, 30	mpbir 234	. . 3 ⊢ (ℝ × ℝ) ∈ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦))
32		elpwuni 5065	. . 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 233	1 ⊢ ∪ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867   × cxp 5649  ran crn 5652  ‘cfv 6531   ∈ cmpo 7414  ℝcr 11180  sigAlgebracsiga 34722  𝔅_ℝcbrsiga 34796

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-pre-lttri 11255  ax-pre-lttrn 11256

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-ioo 13461  df-topgen 17594  df-top 23192  df-bases 23244  df-siga 34723  df-sigagen 34754  df-brsiga 34797

This theorem is used by:  sxbrsigalem5  34903  sxbrsiga  34905