Theorem intsaluni 46349

Description: The union of an arbitrary intersection of sigma-algebras on the same set 𝑋, is 𝑋. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
intsaluni.ga	⊢ (𝜑 → 𝐺 ⊆ SAlg)
intsaluni.gn0	⊢ (𝜑 → 𝐺 ≠ ∅)
intsaluni.x	⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 = 𝑋)

Assertion

Ref	Expression
intsaluni	⊢ (𝜑 → ∪ ∩ 𝐺 = 𝑋)

Distinct variable groups:   𝐺,𝑠   𝑋,𝑠   𝜑,𝑠

Proof of Theorem intsaluni

Dummy variables 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1913	. 2 ⊢ Ⅎ𝑠𝜑
2		nfv 1913	. 2 ⊢ Ⅎ𝑠∪ ∩ 𝐺 = 𝑋
3		intsaluni.gn0	. . 3 ⊢ (𝜑 → 𝐺 ≠ ∅)
4		n0 4352	. . . 4 ⊢ (𝐺 ≠ ∅ ↔ ∃𝑠 𝑠 ∈ 𝐺)
5	4	biimpi 216	. . 3 ⊢ (𝐺 ≠ ∅ → ∃𝑠 𝑠 ∈ 𝐺)
6	3, 5	syl 17	. 2 ⊢ (𝜑 → ∃𝑠 𝑠 ∈ 𝐺)
7		intss1 4962	. . . . . . 7 ⊢ (𝑠 ∈ 𝐺 → ∩ 𝐺 ⊆ 𝑠)
8	7	unissd 4916	. . . . . 6 ⊢ (𝑠 ∈ 𝐺 → ∪ ∩ 𝐺 ⊆ ∪ 𝑠)
9	8	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ ∩ 𝐺 ⊆ ∪ 𝑠)
10		intsaluni.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 = 𝑋)
11	9, 10	sseqtrd 4019	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ ∩ 𝐺 ⊆ 𝑋)
12	10	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑠 = 𝑋)
13		eleq1w 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ 𝐺 ↔ 𝑡 ∈ 𝐺))
14	13	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → ((𝜑 ∧ 𝑠 ∈ 𝐺) ↔ (𝜑 ∧ 𝑡 ∈ 𝐺)))
15		unieq 4917	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑡 → ∪ 𝑠 = ∪ 𝑡)
16	15	eqeq1d 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → (∪ 𝑠 = 𝑋 ↔ ∪ 𝑡 = 𝑋))
17	14, 16	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 = 𝑋) ↔ ((𝜑 ∧ 𝑡 ∈ 𝐺) → ∪ 𝑡 = 𝑋)))
18	17, 10	chvarvv 1997	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → ∪ 𝑡 = 𝑋)
19	18	eqcomd 2742	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → 𝑋 = ∪ 𝑡)
20	19	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → 𝑋 = ∪ 𝑡)
21	12, 20	eqtrd 2776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑠 = ∪ 𝑡)
22		intsaluni.ga	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ⊆ SAlg)
23	22	sselda 3982	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → 𝑡 ∈ SAlg)
24		saluni 46345	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ SAlg → ∪ 𝑡 ∈ 𝑡)
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → ∪ 𝑡 ∈ 𝑡)
26	25	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑡 ∈ 𝑡)
27	21, 26	eqeltrd 2840	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑠 ∈ 𝑡)
28	27	ralrimiva 3145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∀𝑡 ∈ 𝐺 ∪ 𝑠 ∈ 𝑡)
29		uniexg 7761	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐺 → ∪ 𝑠 ∈ V)
30	29	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 ∈ V)
31		elintg 4953	. . . . . . . . 9 ⊢ (∪ 𝑠 ∈ V → (∪ 𝑠 ∈ ∩ 𝐺 ↔ ∀𝑡 ∈ 𝐺 ∪ 𝑠 ∈ 𝑡))
32	30, 31	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → (∪ 𝑠 ∈ ∩ 𝐺 ↔ ∀𝑡 ∈ 𝐺 ∪ 𝑠 ∈ 𝑡))
33	28, 32	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 ∈ ∩ 𝐺)
34	33	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → ∪ 𝑠 ∈ ∩ 𝐺)
35		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
36	10	eqcomd 2742	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → 𝑋 = ∪ 𝑠)
37	36	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑋 = ∪ 𝑠)
38	35, 37	eleqtrd 2842	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ∪ 𝑠)
39		eleq2 2829	. . . . . . 7 ⊢ (𝑦 = ∪ 𝑠 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ∪ 𝑠))
40	39	rspcev 3621	. . . . . 6 ⊢ ((∪ 𝑠 ∈ ∩ 𝐺 ∧ 𝑥 ∈ ∪ 𝑠) → ∃𝑦 ∈ ∩ 𝐺𝑥 ∈ 𝑦)
41	34, 38, 40	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ ∩ 𝐺𝑥 ∈ 𝑦)
42		eluni2 4910	. . . . 5 ⊢ (𝑥 ∈ ∪ ∩ 𝐺 ↔ ∃𝑦 ∈ ∩ 𝐺𝑥 ∈ 𝑦)
43	41, 42	sylibr 234	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ∪ ∩ 𝐺)
44	11, 43	eqelssd 4004	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ ∩ 𝐺 = 𝑋)
45	44	ex 412	. 2 ⊢ (𝜑 → (𝑠 ∈ 𝐺 → ∪ ∩ 𝐺 = 𝑋))
46	1, 2, 6, 45	exlimimdd 2218	1 ⊢ (𝜑 → ∪ ∩ 𝐺 = 𝑋)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ⊆ wss 3950  ∅c0 4332  ∪ cuni 4906  ∩ cint 4945  SAlgcsalg 46328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-ss 3967  df-nul 4333  df-pw 4601  df-uni 4907  df-int 4946  df-salg 46329

This theorem is referenced by:  intsal  46350  salgenuni  46357