Theorem intsaluni 44112

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 1916	. 2 ⊢ Ⅎ𝑠𝜑
2		nfv 1916	. 2 ⊢ Ⅎ𝑠∪ ∩ 𝐺 = 𝑋
3		intsaluni.gn0	. . 3 ⊢ (𝜑 → 𝐺 ≠ ∅)
4		n0 4290	. . . 4 ⊢ (𝐺 ≠ ∅ ↔ ∃𝑠 𝑠 ∈ 𝐺)
5	4	biimpi 215	. . 3 ⊢ (𝐺 ≠ ∅ → ∃𝑠 𝑠 ∈ 𝐺)
6	3, 5	syl 17	. 2 ⊢ (𝜑 → ∃𝑠 𝑠 ∈ 𝐺)
7		intss1 4905	. . . . . . 7 ⊢ (𝑠 ∈ 𝐺 → ∩ 𝐺 ⊆ 𝑠)
8	7	unissd 4858	. . . . . 6 ⊢ (𝑠 ∈ 𝐺 → ∪ ∩ 𝐺 ⊆ ∪ 𝑠)
9	8	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ ∩ 𝐺 ⊆ ∪ 𝑠)
10		intsaluni.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 = 𝑋)
11	9, 10	sseqtrd 3970	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ ∩ 𝐺 ⊆ 𝑋)
12	10	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑠 = 𝑋)
13		eleq1w 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ 𝐺 ↔ 𝑡 ∈ 𝐺))
14	13	anbi2d 629	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → ((𝜑 ∧ 𝑠 ∈ 𝐺) ↔ (𝜑 ∧ 𝑡 ∈ 𝐺)))
15		unieq 4859	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑡 → ∪ 𝑠 = ∪ 𝑡)
16	15	eqeq1d 2739	. . . . . . . . . . . . . . 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 2001	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → ∪ 𝑡 = 𝑋)
19	18	eqcomd 2743	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → 𝑋 = ∪ 𝑡)
20	19	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → 𝑋 = ∪ 𝑡)
21	12, 20	eqtrd 2777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑠 = ∪ 𝑡)
22		intsaluni.ga	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ⊆ SAlg)
23	22	sselda 3930	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → 𝑡 ∈ SAlg)
24		saluni 44109	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ SAlg → ∪ 𝑡 ∈ 𝑡)
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → ∪ 𝑡 ∈ 𝑡)
26	25	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑡 ∈ 𝑡)
27	21, 26	eqeltrd 2838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑠 ∈ 𝑡)
28	27	ralrimiva 3140	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∀𝑡 ∈ 𝐺 ∪ 𝑠 ∈ 𝑡)
29		uniexg 7631	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐺 → ∪ 𝑠 ∈ V)
30	29	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 ∈ V)
31		elintg 4898	. . . . . . . . 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 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 ∈ ∩ 𝐺)
34	33	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → ∪ 𝑠 ∈ ∩ 𝐺)
35		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
36	10	eqcomd 2743	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → 𝑋 = ∪ 𝑠)
37	36	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑋 = ∪ 𝑠)
38	35, 37	eleqtrd 2840	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ∪ 𝑠)
39		eleq2 2826	. . . . . . 7 ⊢ (𝑦 = ∪ 𝑠 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ∪ 𝑠))
40	39	rspcev 3570	. . . . . 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 4852	. . . . 5 ⊢ (𝑥 ∈ ∪ ∩ 𝐺 ↔ ∃𝑦 ∈ ∩ 𝐺𝑥 ∈ 𝑦)
43	41, 42	sylibr 233	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ∪ ∩ 𝐺)
44	11, 43	eqelssd 3951	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ ∩ 𝐺 = 𝑋)
45	44	ex 413	. 2 ⊢ (𝜑 → (𝑠 ∈ 𝐺 → ∪ ∩ 𝐺 = 𝑋))
46	1, 2, 6, 45	exlimimdd 2211	1 ⊢ (𝜑 → ∪ ∩ 𝐺 = 𝑋)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  Vcvv 3441   ⊆ wss 3896  ∅c0 4266  ∪ cuni 4848  ∩ cint 4890  SAlgcsalg 44093

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3899  df-in 3903  df-ss 3913  df-nul 4267  df-pw 4545  df-uni 4849  df-int 4891  df-salg 44094

This theorem is referenced by:  intsal  44113  salgenuni  44120