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Theorem issiga 34113
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 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
Distinct variable groups:   𝑥,𝑂   𝑥,𝑆

Proof of Theorem issiga
Dummy variables 𝑜 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6944 . . . 4 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
2 elex 3501 . . . 4 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ V)
31, 2jca 511 . . 3 (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V))
43a1i 11 . 2 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) → (𝑂 ∈ V ∧ 𝑆 ∈ V)))
5 simpr1 1195 . . . . 5 ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂𝑆)
6 elex 3501 . . . . 5 (𝑂𝑆𝑂 ∈ V)
75, 6syl 17 . . . 4 ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂 ∈ V)
87a1i 11 . . 3 (𝑆 ∈ V → ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂 ∈ V))
98anc2ri 556 . 2 (𝑆 ∈ V → ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → (𝑂 ∈ V ∧ 𝑆 ∈ V)))
10 df-siga 34110 . . . 4 sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})
11 sigaex 34111 . . . 4 {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V
12 pweq 4614 . . . . . . 7 (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
1312sseq2d 4016 . . . . . 6 (𝑜 = 𝑂 → (𝑠 ⊆ 𝒫 𝑜𝑠 ⊆ 𝒫 𝑂))
14 sseq1 4009 . . . . . 6 (𝑠 = 𝑆 → (𝑠 ⊆ 𝒫 𝑂𝑆 ⊆ 𝒫 𝑂))
1513, 14sylan9bb 509 . . . . 5 ((𝑜 = 𝑂𝑠 = 𝑆) → (𝑠 ⊆ 𝒫 𝑜𝑆 ⊆ 𝒫 𝑂))
16 eleq12 2831 . . . . . 6 ((𝑜 = 𝑂𝑠 = 𝑆) → (𝑜𝑠𝑂𝑆))
17 simpr 484 . . . . . . 7 ((𝑜 = 𝑂𝑠 = 𝑆) → 𝑠 = 𝑆)
18 difeq1 4119 . . . . . . . . . 10 (𝑜 = 𝑂 → (𝑜𝑥) = (𝑂𝑥))
1918adantr 480 . . . . . . . . 9 ((𝑜 = 𝑂𝑠 = 𝑆) → (𝑜𝑥) = (𝑂𝑥))
2019eleq1d 2826 . . . . . . . 8 ((𝑜 = 𝑂𝑠 = 𝑆) → ((𝑜𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑠))
21 eleq2 2830 . . . . . . . . 9 (𝑠 = 𝑆 → ((𝑂𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑆))
2221adantl 481 . . . . . . . 8 ((𝑜 = 𝑂𝑠 = 𝑆) → ((𝑂𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑆))
2320, 22bitrd 279 . . . . . . 7 ((𝑜 = 𝑂𝑠 = 𝑆) → ((𝑜𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑆))
2417, 23raleqbidv 3346 . . . . . 6 ((𝑜 = 𝑂𝑠 = 𝑆) → (∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ↔ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆))
25 pweq 4614 . . . . . . . 8 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
26 eleq2 2830 . . . . . . . . 9 (𝑠 = 𝑆 → ( 𝑥𝑠 𝑥𝑆))
2726imbi2d 340 . . . . . . . 8 (𝑠 = 𝑆 → ((𝑥 ≼ ω → 𝑥𝑠) ↔ (𝑥 ≼ ω → 𝑥𝑆)))
2825, 27raleqbidv 3346 . . . . . . 7 (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
2928adantl 481 . . . . . 6 ((𝑜 = 𝑂𝑠 = 𝑆) → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
3016, 24, 293anbi123d 1438 . . . . 5 ((𝑜 = 𝑂𝑠 = 𝑆) → ((𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)) ↔ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
3115, 30anbi12d 632 . . . 4 ((𝑜 = 𝑂𝑠 = 𝑆) → ((𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
3210, 11, 31abfmpel 32665 . . 3 ((𝑂 ∈ V ∧ 𝑆 ∈ V) → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
3332a1i 11 . 2 (𝑆 ∈ V → ((𝑂 ∈ V ∧ 𝑆 ∈ V) → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))))
344, 9, 33pm5.21ndd 379 1 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cdif 3948  wss 3951  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  cfv 6561  ωcom 7887  cdom 8983  sigAlgebracsiga 34109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-siga 34110
This theorem is referenced by:  baselsiga  34116  sigasspw  34117  issgon  34124  isrnsigau  34128  dmvlsiga  34130  pwsiga  34131  prsiga  34132  sigainb  34137  insiga  34138  sigapildsys  34163  imambfm  34264  carsgsiga  34324
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