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Theorem saliunclf 46337
Description: SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
Hypotheses
Ref Expression
saliunclf.1 𝑘𝜑
saliunclf.2 𝑘𝑆
saliunclf.3 𝑘𝐾
saliunclf.4 (𝜑𝑆 ∈ SAlg)
saliunclf.5 (𝜑𝐾 ≼ ω)
saliunclf.6 ((𝜑𝑘𝐾) → 𝐸𝑆)
Assertion
Ref Expression
saliunclf (𝜑 𝑘𝐾 𝐸𝑆)

Proof of Theorem saliunclf
StepHypRef Expression
1 saliunclf.1 . . . 4 𝑘𝜑
2 saliunclf.6 . . . 4 ((𝜑𝑘𝐾) → 𝐸𝑆)
31, 2ralrimia 3258 . . 3 (𝜑 → ∀𝑘𝐾 𝐸𝑆)
4 dfiun3g 5978 . . 3 (∀𝑘𝐾 𝐸𝑆 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
53, 4syl 17 . 2 (𝜑 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
6 saliunclf.4 . . 3 (𝜑𝑆 ∈ SAlg)
7 saliunclf.3 . . . . 5 𝑘𝐾
8 saliunclf.2 . . . . 5 𝑘𝑆
9 eqid 2737 . . . . 5 (𝑘𝐾𝐸) = (𝑘𝐾𝐸)
101, 7, 8, 9, 2rnmptssdff 45282 . . . 4 (𝜑 → ran (𝑘𝐾𝐸) ⊆ 𝑆)
116, 10sselpwd 5328 . . 3 (𝜑 → ran (𝑘𝐾𝐸) ∈ 𝒫 𝑆)
12 saliunclf.5 . . . 4 (𝜑𝐾 ≼ ω)
137rn1st 45280 . . . 4 (𝐾 ≼ ω → ran (𝑘𝐾𝐸) ≼ ω)
1412, 13syl 17 . . 3 (𝜑 → ran (𝑘𝐾𝐸) ≼ ω)
156, 11, 14salunicl 46331 . 2 (𝜑 ran (𝑘𝐾𝐸) ∈ 𝑆)
165, 15eqeltrd 2841 1 (𝜑 𝑘𝐾 𝐸𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wnf 1783  wcel 2108  wnfc 2890  wral 3061   cuni 4907   ciun 4991   class class class wbr 5143  cmpt 5225  ran crn 5686  ωcom 7887  cdom 8983  SAlgcsalg 46323
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
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-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-card 9979  df-acn 9982  df-salg 46324
This theorem is referenced by:  saliuncl  46338  saliinclf  46341  smfsupdmmbllem  46859  smfinfdmmbllem  46863
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