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Theorem isrnsiga 32752
Description: The property of being a sigma-algebra on an indefinite base set. (Contributed by Thierry Arnoux, 3-Sep-2016.) (Proof shortened by Thierry Arnoux, 23-Oct-2016.)
Assertion
Ref Expression
isrnsiga (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
Distinct variable group:   𝑥,𝑜,𝑆

Proof of Theorem isrnsiga
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 df-siga 32748 . . 3 sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})
2 sigaex 32749 . . 3 {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V
3 sseq1 3974 . . . 4 (𝑠 = 𝑆 → (𝑠 ⊆ 𝒫 𝑜𝑆 ⊆ 𝒫 𝑜))
4 eleq2 2827 . . . . 5 (𝑠 = 𝑆 → (𝑜𝑠𝑜𝑆))
5 eleq2 2827 . . . . . 6 (𝑠 = 𝑆 → ((𝑜𝑥) ∈ 𝑠 ↔ (𝑜𝑥) ∈ 𝑆))
65raleqbi1dv 3310 . . . . 5 (𝑠 = 𝑆 → (∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ↔ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆))
7 pweq 4579 . . . . . 6 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
8 eleq2 2827 . . . . . . 7 (𝑠 = 𝑆 → ( 𝑥𝑠 𝑥𝑆))
98imbi2d 341 . . . . . 6 (𝑠 = 𝑆 → ((𝑥 ≼ ω → 𝑥𝑠) ↔ (𝑥 ≼ ω → 𝑥𝑆)))
107, 9raleqbidv 3322 . . . . 5 (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
114, 6, 103anbi123d 1437 . . . 4 (𝑠 = 𝑆 → ((𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)) ↔ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
123, 11anbi12d 632 . . 3 (𝑠 = 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)))))
131, 2, 12abfmpunirn 31610 . 2 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜 ∈ V (𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
14 rexv 3473 . . 3 (∃𝑜 ∈ V (𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) ↔ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
1514anbi2i 624 . 2 ((𝑆 ∈ V ∧ ∃𝑜 ∈ V (𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))) ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
1613, 15bitri 275 1 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃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 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wral 3065  wrex 3074  Vcvv 3448  cdif 3912  wss 3915  𝒫 cpw 4565   cuni 4870   class class class wbr 5110  ran crn 5639  ωcom 7807  cdom 8888  sigAlgebracsiga 32747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509  df-siga 32748
This theorem is referenced by:  0elsiga  32753  sigaclcu  32756  issgon  32762
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