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Theorem sigaclcuni 34119
Description: A sigma-algebra is closed under countable union: indexed union version. (Contributed by Thierry Arnoux, 8-Jun-2017.)
Hypothesis
Ref Expression
sigaclcuni.1 𝑘𝐴
Assertion
Ref Expression
sigaclcuni ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑘𝐴 𝐵𝑆)
Distinct variable group:   𝑆,𝑘
Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)

Proof of Theorem sigaclcuni
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfiun2g 5030 . . 3 (∀𝑘𝐴 𝐵𝑆 𝑘𝐴 𝐵 = {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵})
213ad2ant2 1135 . 2 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑘𝐴 𝐵 = {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵})
3 simp1 1137 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑆 ran sigAlgebra)
4 r19.29 3114 . . . . . . . 8 ((∀𝑘𝐴 𝐵𝑆 ∧ ∃𝑘𝐴 𝑧 = 𝐵) → ∃𝑘𝐴 (𝐵𝑆𝑧 = 𝐵))
5 simpr 484 . . . . . . . . . 10 ((𝐵𝑆𝑧 = 𝐵) → 𝑧 = 𝐵)
6 simpl 482 . . . . . . . . . 10 ((𝐵𝑆𝑧 = 𝐵) → 𝐵𝑆)
75, 6eqeltrd 2841 . . . . . . . . 9 ((𝐵𝑆𝑧 = 𝐵) → 𝑧𝑆)
87rexlimivw 3151 . . . . . . . 8 (∃𝑘𝐴 (𝐵𝑆𝑧 = 𝐵) → 𝑧𝑆)
94, 8syl 17 . . . . . . 7 ((∀𝑘𝐴 𝐵𝑆 ∧ ∃𝑘𝐴 𝑧 = 𝐵) → 𝑧𝑆)
109ex 412 . . . . . 6 (∀𝑘𝐴 𝐵𝑆 → (∃𝑘𝐴 𝑧 = 𝐵𝑧𝑆))
1110abssdv 4068 . . . . 5 (∀𝑘𝐴 𝐵𝑆 → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆)
12113ad2ant2 1135 . . . 4 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆)
13 elpw2g 5333 . . . . 5 (𝑆 ran sigAlgebra → ({𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆 ↔ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆))
143, 13syl 17 . . . 4 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → ({𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆 ↔ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆))
1512, 14mpbird 257 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆)
16 sigaclcuni.1 . . . . 5 𝑘𝐴
1716abrexctf 32730 . . . 4 (𝐴 ≼ ω → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω)
18173ad2ant3 1136 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω)
19 sigaclcu 34118 . . 3 ((𝑆 ran sigAlgebra ∧ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆 ∧ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝑆)
203, 15, 18, 19syl3anc 1373 . 2 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝑆)
212, 20eqeltrd 2841 1 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑘𝐴 𝐵𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wnfc 2890  wral 3061  wrex 3070  wss 3951  𝒫 cpw 4600   cuni 4907   ciun 4991   class class class wbr 5143  ran crn 5686  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-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  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-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-card 9979  df-acn 9982  df-ac 10156  df-siga 34110
This theorem is referenced by:  measvuni  34215  imambfm  34264  sibfof  34342
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