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Theorem saliuncl 41326
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 3175 . . 3 (𝜑 → ∀𝑘𝐾 𝐸𝑆)
3 dfiun3g 5611 . . 3 (∀𝑘𝐾 𝐸𝑆 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
42, 3syl 17 . 2 (𝜑 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
5 saliuncl.s . . 3 (𝜑𝑆 ∈ SAlg)
6 eqid 2825 . . . . . 6 (𝑘𝐾𝐸) = (𝑘𝐾𝐸)
76rnmptss 6641 . . . . 5 (∀𝑘𝐾 𝐸𝑆 → ran (𝑘𝐾𝐸) ⊆ 𝑆)
82, 7syl 17 . . . 4 (𝜑 → ran (𝑘𝐾𝐸) ⊆ 𝑆)
95, 8ssexd 5030 . . . . 5 (𝜑 → ran (𝑘𝐾𝐸) ∈ V)
10 elpwg 4386 . . . . 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 249 . . 3 (𝜑 → ran (𝑘𝐾𝐸) ∈ 𝒫 𝑆)
13 saliuncl.kct . . . 4 (𝜑𝐾 ≼ ω)
14 1stcrestlem 21626 . . . 4 (𝐾 ≼ ω → ran (𝑘𝐾𝐸) ≼ ω)
1513, 14syl 17 . . 3 (𝜑 → ran (𝑘𝐾𝐸) ≼ ω)
165, 12, 15salunicl 41320 . 2 (𝜑 ran (𝑘𝐾𝐸) ∈ 𝑆)
174, 16eqeltrd 2906 1 (𝜑 𝑘𝐾 𝐸𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  wral 3117  Vcvv 3414  wss 3798  𝒫 cpw 4378   cuni 4658   ciun 4740   class class class wbr 4873  cmpt 4952  ran crn 5343  ωcom 7326  cdom 8220  SAlgcsalg 41312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-er 8009  df-map 8124  df-en 8223  df-dom 8224  df-card 9078  df-acn 9081  df-salg 41313
This theorem is referenced by:  saliincl  41329  subsaliuncl  41360  meaiunlelem  41469  meaiuninclem  41481  meaiuninc3v  41485  meaiininclem  41487  caratheodory  41529  opnvonmbllem2  41634  ctvonmbl  41690  vonct  41694  smfaddlem2  41759  smflimlem1  41766  smfresal  41782  smfmullem4  41788
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