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Theorem sigaclcuni 30502
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 4744 . . 3 (∀𝑘𝐴 𝐵𝑆 𝑘𝐴 𝐵 = {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵})
213ad2ant2 1157 . 2 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑘𝐴 𝐵 = {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵})
3 simp1 1159 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑆 ran sigAlgebra)
4 r19.29 3260 . . . . . . . 8 ((∀𝑘𝐴 𝐵𝑆 ∧ ∃𝑘𝐴 𝑧 = 𝐵) → ∃𝑘𝐴 (𝐵𝑆𝑧 = 𝐵))
5 simpr 473 . . . . . . . . . 10 ((𝐵𝑆𝑧 = 𝐵) → 𝑧 = 𝐵)
6 simpl 470 . . . . . . . . . 10 ((𝐵𝑆𝑧 = 𝐵) → 𝐵𝑆)
75, 6eqeltrd 2885 . . . . . . . . 9 ((𝐵𝑆𝑧 = 𝐵) → 𝑧𝑆)
87rexlimivw 3217 . . . . . . . 8 (∃𝑘𝐴 (𝐵𝑆𝑧 = 𝐵) → 𝑧𝑆)
94, 8syl 17 . . . . . . 7 ((∀𝑘𝐴 𝐵𝑆 ∧ ∃𝑘𝐴 𝑧 = 𝐵) → 𝑧𝑆)
109ex 399 . . . . . 6 (∀𝑘𝐴 𝐵𝑆 → (∃𝑘𝐴 𝑧 = 𝐵𝑧𝑆))
1110abssdv 3873 . . . . 5 (∀𝑘𝐴 𝐵𝑆 → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆)
12113ad2ant2 1157 . . . 4 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆)
13 elpw2g 5019 . . . . 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 248 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆)
16 sigaclcuni.1 . . . . 5 𝑘𝐴
1716abrexctf 29819 . . . 4 (𝐴 ≼ ω → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω)
18173ad2ant3 1158 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω)
19 sigaclcu 30501 . . 3 ((𝑆 ran sigAlgebra ∧ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆 ∧ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝑆)
203, 15, 18, 19syl3anc 1483 . 2 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝑆)
212, 20eqeltrd 2885 1 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑘𝐴 𝐵𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  {cab 2792  wnfc 2935  wral 3096  wrex 3097  wss 3769  𝒫 cpw 4351   cuni 4630   ciun 4712   class class class wbr 4844  ran crn 5312  ωcom 7291  cdom 8186  sigAlgebracsiga 30491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-inf2 8781  ax-ac2 9566
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-isom 6106  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1st 7394  df-2nd 7395  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-oadd 7796  df-er 7975  df-map 8090  df-en 8189  df-dom 8190  df-sdom 8191  df-fin 8192  df-oi 8650  df-card 9044  df-acn 9047  df-ac 9218  df-siga 30492
This theorem is referenced by:  measvuni  30598  imambfm  30645  sibfof  30723
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