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Theorem sigaclcuni 30514
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 4686 . . 3 (∀𝑘𝐴 𝐵𝑆 𝑘𝐴 𝐵 = {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵})
213ad2ant2 1128 . 2 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑘𝐴 𝐵 = {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵})
3 simp1 1130 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑆 ran sigAlgebra)
4 r19.29 3220 . . . . . . . 8 ((∀𝑘𝐴 𝐵𝑆 ∧ ∃𝑘𝐴 𝑧 = 𝐵) → ∃𝑘𝐴 (𝐵𝑆𝑧 = 𝐵))
5 simpr 471 . . . . . . . . . 10 ((𝐵𝑆𝑧 = 𝐵) → 𝑧 = 𝐵)
6 simpl 468 . . . . . . . . . 10 ((𝐵𝑆𝑧 = 𝐵) → 𝐵𝑆)
75, 6eqeltrd 2850 . . . . . . . . 9 ((𝐵𝑆𝑧 = 𝐵) → 𝑧𝑆)
87rexlimivw 3177 . . . . . . . 8 (∃𝑘𝐴 (𝐵𝑆𝑧 = 𝐵) → 𝑧𝑆)
94, 8syl 17 . . . . . . 7 ((∀𝑘𝐴 𝐵𝑆 ∧ ∃𝑘𝐴 𝑧 = 𝐵) → 𝑧𝑆)
109ex 397 . . . . . 6 (∀𝑘𝐴 𝐵𝑆 → (∃𝑘𝐴 𝑧 = 𝐵𝑧𝑆))
1110abssdv 3825 . . . . 5 (∀𝑘𝐴 𝐵𝑆 → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆)
12113ad2ant2 1128 . . . 4 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ⊆ 𝑆)
13 elpw2g 4958 . . . . 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 247 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆)
16 sigaclcuni.1 . . . . 5 𝑘𝐴
1716abrexctf 29829 . . . 4 (𝐴 ≼ ω → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω)
18173ad2ant3 1129 . . 3 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω)
19 sigaclcu 30513 . . 3 ((𝑆 ran sigAlgebra ∧ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆 ∧ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝑆)
203, 15, 18, 19syl3anc 1476 . 2 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐵} ∈ 𝑆)
212, 20eqeltrd 2850 1 ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑘𝐴 𝐵𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  {cab 2757  wnfc 2900  wral 3061  wrex 3062  wss 3723  𝒫 cpw 4297   cuni 4574   ciun 4654   class class class wbr 4786  ran crn 5250  ωcom 7210  cdom 8105  sigAlgebracsiga 30503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-inf2 8700  ax-ac2 9485
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-isom 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-map 8009  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-oi 8569  df-card 8963  df-acn 8966  df-ac 9137  df-siga 30504
This theorem is referenced by:  measvuni  30610  imambfm  30657  sibfof  30735
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