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Theorem sigaclcuni 31276
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 4946 . . 3 (∀𝑘𝐴 𝐵𝑆 𝑘𝐴 𝐵 = {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵})
213ad2ant2 1126 . 2 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑘𝐴 𝐵 = {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵})
3 simp1 1128 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑆 ran sigAlgebra)
4 r19.29 3251 . . . . . . . 8 ((∀𝑘𝐴 𝐵𝑆 ∧ ∃𝑘𝐴 𝑧 = 𝐵) → ∃𝑘𝐴 (𝐵𝑆𝑧 = 𝐵))
5 simpr 485 . . . . . . . . . 10 ((𝐵𝑆𝑧 = 𝐵) → 𝑧 = 𝐵)
6 simpl 483 . . . . . . . . . 10 ((𝐵𝑆𝑧 = 𝐵) → 𝐵𝑆)
75, 6eqeltrd 2910 . . . . . . . . 9 ((𝐵𝑆𝑧 = 𝐵) → 𝑧𝑆)
87rexlimivw 3279 . . . . . . . 8 (∃𝑘𝐴 (𝐵𝑆𝑧 = 𝐵) → 𝑧𝑆)
94, 8syl 17 . . . . . . 7 ((∀𝑘𝐴 𝐵𝑆 ∧ ∃𝑘𝐴 𝑧 = 𝐵) → 𝑧𝑆)
109ex 413 . . . . . 6 (∀𝑘𝐴 𝐵𝑆 → (∃𝑘𝐴 𝑧 = 𝐵𝑧𝑆))
1110abssdv 4042 . . . . 5 (∀𝑘𝐴 𝐵𝑆 → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆)
12113ad2ant2 1126 . . . 4 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆)
13 elpw2g 5238 . . . . 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 258 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆)
16 sigaclcuni.1 . . . . 5 𝑘𝐴
1716abrexctf 30380 . . . 4 (𝐴 ≼ ω → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω)
18173ad2ant3 1127 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω)
19 sigaclcu 31275 . . 3 ((𝑆 ran sigAlgebra ∧ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆 ∧ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝑆)
203, 15, 18, 19syl3anc 1363 . 2 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝑆)
212, 20eqeltrd 2910 1 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑘𝐴 𝐵𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wnfc 2958  wral 3135  wrex 3136  wss 3933  𝒫 cpw 4535   cuni 4830   ciun 4910   class class class wbr 5057  ran crn 5549  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-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-ac2 9873
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  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-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-card 9356  df-acn 9359  df-ac 9530  df-siga 31267
This theorem is referenced by:  measvuni  31372  imambfm  31419  sibfof  31497
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