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Theorem saliuncl 41059
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 3115 . . 3 (𝜑 → ∀𝑘𝐾 𝐸𝑆)
3 dfiun3g 5516 . . 3 (∀𝑘𝐾 𝐸𝑆 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
42, 3syl 17 . 2 (𝜑 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
5 saliuncl.s . . 3 (𝜑𝑆 ∈ SAlg)
6 eqid 2771 . . . . . 6 (𝑘𝐾𝐸) = (𝑘𝐾𝐸)
76rnmptss 6534 . . . . 5 (∀𝑘𝐾 𝐸𝑆 → ran (𝑘𝐾𝐸) ⊆ 𝑆)
82, 7syl 17 . . . 4 (𝜑 → ran (𝑘𝐾𝐸) ⊆ 𝑆)
95, 8ssexd 4939 . . . . 5 (𝜑 → ran (𝑘𝐾𝐸) ∈ V)
10 elpwg 4305 . . . . 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 247 . . 3 (𝜑 → ran (𝑘𝐾𝐸) ∈ 𝒫 𝑆)
13 saliuncl.kct . . . 4 (𝜑𝐾 ≼ ω)
14 1stcrestlem 21476 . . . 4 (𝐾 ≼ ω → ran (𝑘𝐾𝐸) ≼ ω)
1513, 14syl 17 . . 3 (𝜑 → ran (𝑘𝐾𝐸) ≼ ω)
165, 12, 15salunicl 41053 . 2 (𝜑 ran (𝑘𝐾𝐸) ∈ 𝑆)
174, 16eqeltrd 2850 1 (𝜑 𝑘𝐾 𝐸𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  wss 3723  𝒫 cpw 4297   cuni 4574   ciun 4654   class class class wbr 4786  cmpt 4863  ran crn 5250  ωcom 7212  cdom 8107  SAlgcsalg 41045
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 7096
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 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-card 8965  df-acn 8968  df-salg 41046
This theorem is referenced by:  saliincl  41062  subsaliuncl  41093  meaiunlelem  41202  meaiuninclem  41214  meaiuninc3v  41218  meaiininclem  41220  caratheodory  41262  opnvonmbllem2  41367  ctvonmbl  41423  vonct  41427  smfaddlem2  41492  smflimlem1  41499  smfresal  41515  smfmullem4  41521
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