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Theorem saliuncl 42894
 Description: SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
saliuncl.s (𝜑𝑆 ∈ SAlg)
saliuncl.kct (𝜑𝐾 ≼ ω)
saliuncl.b ((𝜑𝑘𝐾) → 𝐸𝑆)
Assertion
Ref Expression
saliuncl (𝜑 𝑘𝐾 𝐸𝑆)
Distinct variable groups:   𝑘,𝐾   𝑆,𝑘   𝜑,𝑘
Allowed substitution hint:   𝐸(𝑘)

Proof of Theorem saliuncl
StepHypRef Expression
1 saliuncl.b . . . 4 ((𝜑𝑘𝐾) → 𝐸𝑆)
21ralrimiva 3177 . . 3 (𝜑 → ∀𝑘𝐾 𝐸𝑆)
3 dfiun3g 5822 . . 3 (∀𝑘𝐾 𝐸𝑆 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
42, 3syl 17 . 2 (𝜑 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
5 saliuncl.s . . 3 (𝜑𝑆 ∈ SAlg)
6 eqid 2824 . . . . . 6 (𝑘𝐾𝐸) = (𝑘𝐾𝐸)
76rnmptss 6877 . . . . 5 (∀𝑘𝐾 𝐸𝑆 → ran (𝑘𝐾𝐸) ⊆ 𝑆)
82, 7syl 17 . . . 4 (𝜑 → ran (𝑘𝐾𝐸) ⊆ 𝑆)
95, 8ssexd 5214 . . . . 5 (𝜑 → ran (𝑘𝐾𝐸) ∈ V)
10 elpwg 4525 . . . . 5 (ran (𝑘𝐾𝐸) ∈ V → (ran (𝑘𝐾𝐸) ∈ 𝒫 𝑆 ↔ ran (𝑘𝐾𝐸) ⊆ 𝑆))
119, 10syl 17 . . . 4 (𝜑 → (ran (𝑘𝐾𝐸) ∈ 𝒫 𝑆 ↔ ran (𝑘𝐾𝐸) ⊆ 𝑆))
128, 11mpbird 260 . . 3 (𝜑 → ran (𝑘𝐾𝐸) ∈ 𝒫 𝑆)
13 saliuncl.kct . . . 4 (𝜑𝐾 ≼ ω)
14 1stcrestlem 22063 . . . 4 (𝐾 ≼ ω → ran (𝑘𝐾𝐸) ≼ ω)
1513, 14syl 17 . . 3 (𝜑 → ran (𝑘𝐾𝐸) ≼ ω)
165, 12, 15salunicl 42888 . 2 (𝜑 ran (𝑘𝐾𝐸) ∈ 𝑆)
174, 16eqeltrd 2916 1 (𝜑 𝑘𝐾 𝐸𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133  Vcvv 3480   ⊆ wss 3919  𝒫 cpw 4522  ∪ cuni 4824  ∪ ciun 4905   class class class wbr 5052   ↦ cmpt 5132  ran crn 5543  ωcom 7574   ≼ cdom 8503  SAlgcsalg 42880 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-card 9365  df-acn 9368  df-salg 42881 This theorem is referenced by:  saliincl  42897  subsaliuncl  42928  meaiunlelem  43037  meaiuninclem  43049  meaiuninc3v  43053  meaiininclem  43055  caratheodory  43097  opnvonmbllem2  43202  ctvonmbl  43258  vonct  43262  smfaddlem2  43327  smflimlem1  43334  smfresal  43350  smfmullem4  43356
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