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Theorem sigaclcuni 32757
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 4995 . . 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 3118 . . . . . . . 8 ((∀𝑘𝐴 𝐵𝑆 ∧ ∃𝑘𝐴 𝑧 = 𝐵) → ∃𝑘𝐴 (𝐵𝑆𝑧 = 𝐵))
5 simpr 486 . . . . . . . . . 10 ((𝐵𝑆𝑧 = 𝐵) → 𝑧 = 𝐵)
6 simpl 484 . . . . . . . . . 10 ((𝐵𝑆𝑧 = 𝐵) → 𝐵𝑆)
75, 6eqeltrd 2838 . . . . . . . . 9 ((𝐵𝑆𝑧 = 𝐵) → 𝑧𝑆)
87rexlimivw 3149 . . . . . . . 8 (∃𝑘𝐴 (𝐵𝑆𝑧 = 𝐵) → 𝑧𝑆)
94, 8syl 17 . . . . . . 7 ((∀𝑘𝐴 𝐵𝑆 ∧ ∃𝑘𝐴 𝑧 = 𝐵) → 𝑧𝑆)
109ex 414 . . . . . 6 (∀𝑘𝐴 𝐵𝑆 → (∃𝑘𝐴 𝑧 = 𝐵𝑧𝑆))
1110abssdv 4030 . . . . 5 (∀𝑘𝐴 𝐵𝑆 → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆)
12113ad2ant2 1135 . . . 4 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆)
13 elpw2g 5306 . . . . 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 31677 . . . 4 (𝐴 ≼ ω → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω)
18173ad2ant3 1136 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω)
19 sigaclcu 32756 . . 3 ((𝑆 ran sigAlgebra ∧ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆 ∧ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝑆)
203, 15, 18, 19syl3anc 1372 . 2 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝑆)
212, 20eqeltrd 2838 1 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑘𝐴 𝐵𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2714  wnfc 2888  wral 3065  wrex 3074  wss 3915  𝒫 cpw 4565   cuni 4870   ciun 4959   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-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-ac2 10406
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-rmo 3356  df-reu 3357  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-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  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-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9453  df-card 9882  df-acn 9885  df-ac 10059  df-siga 32748
This theorem is referenced by:  measvuni  32853  imambfm  32902  sibfof  32980
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