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Theorem saliuncl 41466
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 3148 . . 3 (𝜑 → ∀𝑘𝐾 𝐸𝑆)
3 dfiun3g 5624 . . 3 (∀𝑘𝐾 𝐸𝑆 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
42, 3syl 17 . 2 (𝜑 𝑘𝐾 𝐸 = ran (𝑘𝐾𝐸))
5 saliuncl.s . . 3 (𝜑𝑆 ∈ SAlg)
6 eqid 2778 . . . . . 6 (𝑘𝐾𝐸) = (𝑘𝐾𝐸)
76rnmptss 6656 . . . . 5 (∀𝑘𝐾 𝐸𝑆 → ran (𝑘𝐾𝐸) ⊆ 𝑆)
82, 7syl 17 . . . 4 (𝜑 → ran (𝑘𝐾𝐸) ⊆ 𝑆)
95, 8ssexd 5042 . . . . 5 (𝜑 → ran (𝑘𝐾𝐸) ∈ V)
10 elpwg 4387 . . . . 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 21664 . . . 4 (𝐾 ≼ ω → ran (𝑘𝐾𝐸) ≼ ω)
1513, 14syl 17 . . 3 (𝜑 → ran (𝑘𝐾𝐸) ≼ ω)
165, 12, 15salunicl 41460 . 2 (𝜑 ran (𝑘𝐾𝐸) ∈ 𝑆)
174, 16eqeltrd 2859 1 (𝜑 𝑘𝐾 𝐸𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wral 3090  Vcvv 3398  wss 3792  𝒫 cpw 4379   cuni 4671   ciun 4753   class class class wbr 4886  cmpt 4965  ran crn 5356  ωcom 7343  cdom 8239  SAlgcsalg 41452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-card 9098  df-acn 9101  df-salg 41453
This theorem is referenced by:  saliincl  41469  subsaliuncl  41500  meaiunlelem  41609  meaiuninclem  41621  meaiuninc3v  41625  meaiininclem  41627  caratheodory  41669  opnvonmbllem2  41774  ctvonmbl  41830  vonct  41834  smfaddlem2  41899  smflimlem1  41906  smfresal  41922  smfmullem4  41928
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