Theorem issiga 31599

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 6691	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
2		elex 3428	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
3	1, 2	jca 515	. . 3 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V))
4	3	a1i 11	. 2 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V)))
5		simpr1 1191	. . . . 5 ⊢ ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → 𝑂 ∈ 𝑆)
6		elex 3428	. . . . 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 560	. 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 31596	. . . 4 ⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
11		sigaex 31597	. . . 4 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V
12		pweq 4510	. . . . . . 7 ⊢ (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
13	12	sseq2d 3924	. . . . . 6 ⊢ (𝑜 = 𝑂 → (𝑠 ⊆ 𝒫 𝑜 ↔ 𝑠 ⊆ 𝒫 𝑂))
14		sseq1 3917	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠 ⊆ 𝒫 𝑂 ↔ 𝑆 ⊆ 𝒫 𝑂))
15	13, 14	sylan9bb 513	. . . . 5 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑠 ⊆ 𝒫 𝑜 ↔ 𝑆 ⊆ 𝒫 𝑂))
16		eleq12 2841	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑜 ∈ 𝑠 ↔ 𝑂 ∈ 𝑆))
17		simpr 488	. . . . . . 7 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
18		difeq1 4021	. . . . . . . . . 10 ⊢ (𝑜 = 𝑂 → (𝑜 ∖ 𝑥) = (𝑂 ∖ 𝑥))
19	18	adantr 484	. . . . . . . . 9 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (𝑜 ∖ 𝑥) = (𝑂 ∖ 𝑥))
20	19	eleq1d 2836	. . . . . . . 8 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑜 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑠))
21		eleq2 2840	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((𝑂 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
22	21	adantl 485	. . . . . . . 8 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑂 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
23	20, 22	bitrd 282	. . . . . . 7 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → ((𝑜 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
24	17, 23	raleqbidv 3319	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆))
25		pweq 4510	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
26		eleq2 2840	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (∪ 𝑥 ∈ 𝑠 ↔ ∪ 𝑥 ∈ 𝑆))
27	26	imbi2d 344	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
28	25, 27	raleqbidv 3319	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
29	28	adantl 485	. . . . . 6 ⊢ ((𝑜 = 𝑂 ∧ 𝑠 = 𝑆) → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
30	16, 24, 29	3anbi123d 1433	. . . . 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 633	. . . 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 30516	. . 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 384	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ∖ cdif 3855   ⊆ wss 3858  𝒫 cpw 4494  ∪ cuni 4798   class class class wbr 5032  ‘cfv 6335  ωcom 7579   ≼ cdom 8525  sigAlgebracsiga 31595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-siga 31596

This theorem is referenced by:  baselsiga  31602  sigasspw  31603  issgon  31610  isrnsigau  31614  dmvlsiga  31616  pwsiga  31617  prsiga  31618  sigainb  31623  insiga  31624  sigapildsys  31649  imambfm  31748  carsgsiga  31808