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Theorem sigaval 34112
Description: The set of sigma-algebra with a given base set. (Contributed by Thierry Arnoux, 23-Sep-2016.)
Assertion
Ref Expression
sigaval (𝑂 ∈ V → (sigAlgebra‘𝑂) = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})
Distinct variable group:   𝑥,𝑠,𝑂

Proof of Theorem sigaval
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 df-rab 3437 . . . 4 {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))} = {𝑠 ∣ (𝑠 ∈ 𝒫 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))}
2 velpw 4605 . . . . . 6 (𝑠 ∈ 𝒫 𝒫 𝑂𝑠 ⊆ 𝒫 𝑂)
32anbi1i 624 . . . . 5 ((𝑠 ∈ 𝒫 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))))
43abbii 2809 . . . 4 {𝑠 ∣ (𝑠 ∈ 𝒫 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))}
51, 4eqtri 2765 . . 3 {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))} = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))}
6 pwexg 5378 . . . 4 (𝑂 ∈ V → 𝒫 𝑂 ∈ V)
7 pwexg 5378 . . . 4 (𝒫 𝑂 ∈ V → 𝒫 𝒫 𝑂 ∈ V)
8 rabexg 5337 . . . 4 (𝒫 𝒫 𝑂 ∈ V → {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))} ∈ V)
96, 7, 83syl 18 . . 3 (𝑂 ∈ V → {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))} ∈ V)
105, 9eqeltrrid 2846 . 2 (𝑂 ∈ V → {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V)
11 pweq 4614 . . . . . 6 (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
1211sseq2d 4016 . . . . 5 (𝑜 = 𝑂 → (𝑠 ⊆ 𝒫 𝑜𝑠 ⊆ 𝒫 𝑂))
13 eleq1 2829 . . . . . 6 (𝑜 = 𝑂 → (𝑜𝑠𝑂𝑠))
14 difeq1 4119 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜𝑥) = (𝑂𝑥))
1514eleq1d 2826 . . . . . . 7 (𝑜 = 𝑂 → ((𝑜𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑠))
1615ralbidv 3178 . . . . . 6 (𝑜 = 𝑂 → (∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ↔ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠))
1713, 163anbi12d 1439 . . . . 5 (𝑜 = 𝑂 → ((𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)) ↔ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))))
1812, 17anbi12d 632 . . . 4 (𝑜 = 𝑂 → ((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)))))
1918abbidv 2808 . . 3 (𝑜 = 𝑂 → {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})
20 df-siga 34110 . . 3 sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})
2119, 20fvmptg 7014 . 2 ((𝑂 ∈ V ∧ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V) → (sigAlgebra‘𝑂) = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})
2210, 21mpdan 687 1 (𝑂 ∈ V → (sigAlgebra‘𝑂) = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wral 3061  {crab 3436  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-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: (None)
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