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Theorem issiga 34092
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 3498 . . . 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 1193 . . . . 5 ((𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → 𝑂𝑆)
6 elex 3498 . . . . 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 34089 . . . 4 sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})
11 sigaex 34090 . . . 4 {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V
12 pweq 4618 . . . . . . 7 (𝑜 = 𝑂 → 𝒫 𝑜 = 𝒫 𝑂)
1312sseq2d 4027 . . . . . 6 (𝑜 = 𝑂 → (𝑠 ⊆ 𝒫 𝑜𝑠 ⊆ 𝒫 𝑂))
14 sseq1 4020 . . . . . 6 (𝑠 = 𝑆 → (𝑠 ⊆ 𝒫 𝑂𝑆 ⊆ 𝒫 𝑂))
1513, 14sylan9bb 509 . . . . 5 ((𝑜 = 𝑂𝑠 = 𝑆) → (𝑠 ⊆ 𝒫 𝑜𝑆 ⊆ 𝒫 𝑂))
16 eleq12 2828 . . . . . 6 ((𝑜 = 𝑂𝑠 = 𝑆) → (𝑜𝑠𝑂𝑆))
17 simpr 484 . . . . . . 7 ((𝑜 = 𝑂𝑠 = 𝑆) → 𝑠 = 𝑆)
18 difeq1 4128 . . . . . . . . . 10 (𝑜 = 𝑂 → (𝑜𝑥) = (𝑂𝑥))
1918adantr 480 . . . . . . . . 9 ((𝑜 = 𝑂𝑠 = 𝑆) → (𝑜𝑥) = (𝑂𝑥))
2019eleq1d 2823 . . . . . . . 8 ((𝑜 = 𝑂𝑠 = 𝑆) → ((𝑜𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑠))
21 eleq2 2827 . . . . . . . . 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 3343 . . . . . 6 ((𝑜 = 𝑂𝑠 = 𝑆) → (∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ↔ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆))
25 pweq 4618 . . . . . . . 8 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
26 eleq2 2827 . . . . . . . . 9 (𝑠 = 𝑆 → ( 𝑥𝑠 𝑥𝑆))
2726imbi2d 340 . . . . . . . 8 (𝑠 = 𝑆 → ((𝑥 ≼ ω → 𝑥𝑠) ↔ (𝑥 ≼ ω → 𝑥𝑆)))
2825, 27raleqbidv 3343 . . . . . . 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 1435 . . . . 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 32671 . . 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 1086   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  cdif 3959  wss 3962  𝒫 cpw 4604   cuni 4911   class class class wbr 5147  cfv 6562  ωcom 7886  cdom 8981  sigAlgebracsiga 34088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-siga 34089
This theorem is referenced by:  baselsiga  34095  sigasspw  34096  issgon  34103  isrnsigau  34107  dmvlsiga  34109  pwsiga  34110  prsiga  34111  sigainb  34116  insiga  34117  sigapildsys  34142  imambfm  34243  carsgsiga  34303
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