Theorem issiga 32080

Description: An alternative definition of the sigma-algebra, for a given base set. (Contributed by Thierry Arnoux, 19-Sep-2016.)

Assertion

Ref	Expression
issiga	⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))

Distinct variable groups:   𝑥,𝑂   𝑥,𝑆

Proof of Theorem issiga

Dummy variables 𝑜 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6807	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
2		elex 3450	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
3	1, 2	jca 512	. . 3 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V))
4	3	a1i 11	. 2 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V)))
5		simpr1 1193	. . . . 5 ⊢ ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → 𝑂 ∈ 𝑆)
6		elex 3450	. . . . 5 ⊢ (𝑂 ∈ 𝑆 → 𝑂 ∈ V)
7	5, 6	syl 17	. . . 4 ⊢ ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → 𝑂 ∈ V)
8	7	a1i 11	. . 3 ⊢ (𝑆 ∈ V → ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → 𝑂 ∈ V))
9	8	anc2ri 557	. 2 ⊢ (𝑆 ∈ V → ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → (𝑂 ∈ V ∧ 𝑆 ∈ V)))
10		df-siga 32077	. . . 4 ⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
11		sigaex 32078	. . . 4 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V
12		pweq 4549	. . . . . . 7 ⊢ (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
13	12	sseq2d 3953	. . . . . 6 ⊢ (𝑜 = 𝑂 → (𝑠 ⊆ 𝒫 𝑜 ↔ 𝑠 ⊆ 𝒫 𝑂))
14		sseq1 3946	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝒫 𝑂 ↔ 𝑆 ⊆ 𝒫 𝑂))
15	13, 14	sylan9bb 510	. . . . 5 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑠 ⊆ 𝒫 𝑜 ↔ 𝑆 ⊆ 𝒫 𝑂))
16		eleq12 2828	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑜 ∈ 𝑠 ↔ 𝑂 ∈ 𝑆))
17		simpr 485	. . . . . . 7 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
18		difeq1 4050	. . . . . . . . . 10 ⊢ (𝑜 = 𝑂 → (𝑜 ∖ 𝑥) = (𝑂 ∖ 𝑥))
19	18	adantr 481	. . . . . . . . 9 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑜 ∖ 𝑥) = (𝑂 ∖ 𝑥))
20	19	eleq1d 2823	. . . . . . . 8 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑜 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑠))
21		eleq2 2827	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((𝑂 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
22	21	adantl 482	. . . . . . . 8 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑂 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
23	20, 22	bitrd 278	. . . . . . 7 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑜 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
24	17, 23	raleqbidv 3336	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆))
25		pweq 4549	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
26		eleq2 2827	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (∪ 𝑥 ∈ 𝑠 ↔ ∪ 𝑥 ∈ 𝑆))
27	26	imbi2d 341	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
28	25, 27	raleqbidv 3336	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
29	28	adantl 482	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
30	16, 24, 29	3anbi123d 1435	. . . . 5 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)) ↔ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
31	15, 30	anbi12d 631	. . . 4 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
32	10, 11, 31	abfmpel 30992	. . 3 ⊢ ((𝑂 ∈ V ∧ 𝑆 ∈ V) → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
33	32	a1i 11	. 2 ⊢ (𝑆 ∈ V → ((𝑂 ∈ V ∧ 𝑆 ∈ V) → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))))
34	4, 9, 33	pm5.21ndd 381	1 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839   class class class wbr 5074  ‘cfv 6433  ωcom 7712   ≼ cdom 8731  sigAlgebracsiga 32076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-siga 32077

This theorem is referenced by:  baselsiga  32083  sigasspw  32084  issgon  32091  isrnsigau  32095  dmvlsiga  32097  pwsiga  32098  prsiga  32099  sigainb  32104  insiga  32105  sigapildsys  32130  imambfm  32229  carsgsiga  32289