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Theorem saliuncl 39218
 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 2949 . . 3 (𝜑 → ∀𝑘𝐾 𝐸𝑆)
3 dfiun3g 5299 . . 3 (∀𝑘𝐾 𝐸𝑆 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
42, 3syl 17 . 2 (𝜑 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
5 saliuncl.s . . 3 (𝜑𝑆 ∈ SAlg)
6 eqid 2610 . . . . . 6 (𝑘𝐾𝐸) = (𝑘𝐾𝐸)
76rnmptss 6299 . . . . 5 (∀𝑘𝐾 𝐸𝑆 → ran (𝑘𝐾𝐸) ⊆ 𝑆)
82, 7syl 17 . . . 4 (𝜑 → ran (𝑘𝐾𝐸) ⊆ 𝑆)
95, 8ssexd 4733 . . . . 5 (𝜑 → ran (𝑘𝐾𝐸) ∈ V)
10 elpwg 4116 . . . . 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 246 . . 3 (𝜑 → ran (𝑘𝐾𝐸) ∈ 𝒫 𝑆)
13 saliuncl.kct . . . 4 (𝜑𝐾 ≼ ω)
14 1stcrestlem 21065 . . . 4 (𝐾 ≼ ω → ran (𝑘𝐾𝐸) ≼ ω)
1513, 14syl 17 . . 3 (𝜑 → ran (𝑘𝐾𝐸) ≼ ω)
165, 12, 15salunicl 39212 . 2 (𝜑 ran (𝑘𝐾𝐸) ∈ 𝑆)
174, 16eqeltrd 2688 1 (𝜑 𝑘𝐾 𝐸𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ωcom 6957   ≼ cdom 7839  SAlgcsalg 39204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-card 8648  df-acn 8651  df-salg 39205 This theorem is referenced by:  saliincl  39221  subsaliuncl  39252  meaiunlelem  39361  meaiuninclem  39373  meaiininclem  39376  caratheodory  39418  opnvonmbllem2  39523  ctvonmbl  39580  vonct  39584  smfaddlem2  39650  smflimlem1  39657  smfresal  39673  smfmullem4  39679
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