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Theorem sigaval 31269
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 3144 . . . 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 4543 . . . . . 6 (𝑠 ∈ 𝒫 𝒫 𝑂𝑠 ⊆ 𝒫 𝑂)
32anbi1i 623 . . . . 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 2883 . . . 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 2841 . . 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 5270 . . . 4 (𝑂 ∈ V → 𝒫 𝑂 ∈ V)
7 pwexg 5270 . . . 4 (𝒫 𝑂 ∈ V → 𝒫 𝒫 𝑂 ∈ V)
8 rabexg 5225 . . . 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 2915 . 2 (𝑂 ∈ V → {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V)
11 pweq 4538 . . . . . 6 (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
1211sseq2d 3996 . . . . 5 (𝑜 = 𝑂 → (𝑠 ⊆ 𝒫 𝑜𝑠 ⊆ 𝒫 𝑂))
13 eleq1 2897 . . . . . 6 (𝑜 = 𝑂 → (𝑜𝑠𝑂𝑠))
14 difeq1 4089 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜𝑥) = (𝑂𝑥))
1514eleq1d 2894 . . . . . . 7 (𝑜 = 𝑂 → ((𝑜𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑠))
1615ralbidv 3194 . . . . . 6 (𝑜 = 𝑂 → (∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ↔ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠))
1713, 163anbi12d 1428 . . . . 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 630 . . . 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 2882 . . 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 31267 . . 3 sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})
2119, 20fvmptg 6759 . 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 683 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 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wral 3135  {crab 3139  Vcvv 3492  cdif 3930  wss 3933  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  cfv 6348  ωcom 7569  cdom 8495  sigAlgebracsiga 31266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-siga 31267
This theorem is referenced by: (None)
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