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Theorem sigaclci 32100
Description: A sigma-algebra is closed under countable intersections. Deduction version. (Contributed by Thierry Arnoux, 19-Sep-2016.)
Assertion
Ref Expression
sigaclci (((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → 𝐴𝑆)

Proof of Theorem sigaclci
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isrnsigau 32095 . . . . . . . 8 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
21simprd 496 . . . . . . 7 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
32simp2d 1142 . . . . . 6 (𝑆 ran sigAlgebra → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
43adantr 481 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
5 elpwi 4542 . . . . . . . . . . . 12 (𝐴 ∈ 𝒫 𝑆𝐴𝑆)
6 ssrexv 3988 . . . . . . . . . . . 12 (𝐴𝑆 → (∃𝑧𝐴 𝑦 = ( 𝑆𝑧) → ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)))
75, 6syl 17 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 𝑆 → (∃𝑧𝐴 𝑦 = ( 𝑆𝑧) → ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)))
87ss2abdv 3997 . . . . . . . . . 10 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ {𝑦 ∣ ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)})
9 isrnsigau 32095 . . . . . . . . . . . . 13 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → 𝑧𝑆))))
109simprd 496 . . . . . . . . . . . 12 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → 𝑧𝑆)))
1110simp2d 1142 . . . . . . . . . . 11 (𝑆 ran sigAlgebra → ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆)
12 uniiunlem 4019 . . . . . . . . . . . 12 (∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 → (∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 ↔ {𝑦 ∣ ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆))
1311, 12syl 17 . . . . . . . . . . 11 (𝑆 ran sigAlgebra → (∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 ↔ {𝑦 ∣ ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆))
1411, 13mpbid 231 . . . . . . . . . 10 (𝑆 ran sigAlgebra → {𝑦 ∣ ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆)
158, 14sylan9ssr 3935 . . . . . . . . 9 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆)
16 abrexexg 7803 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ V)
17 elpwg 4536 . . . . . . . . . . 11 ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ V → ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆 ↔ {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆))
1816, 17syl 17 . . . . . . . . . 10 (𝐴 ∈ 𝒫 𝑆 → ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆 ↔ {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆))
1918adantl 482 . . . . . . . . 9 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆 ↔ {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆))
2015, 19mpbird 256 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆)
212simp3d 1143 . . . . . . . . 9 (𝑆 ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
2221adantr 481 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
2320, 22jca 512 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
24 abrexdom2jm 30853 . . . . . . . . . 10 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ 𝐴)
25 domtr 8793 . . . . . . . . . 10 (({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ 𝐴𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω)
2624, 25sylan 580 . . . . . . . . 9 ((𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω)
2726ex 413 . . . . . . . 8 (𝐴 ∈ 𝒫 𝑆 → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω))
2827adantl 482 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω))
29 breq1 5077 . . . . . . . . 9 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → (𝑥 ≼ ω ↔ {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω))
30 unieq 4850 . . . . . . . . . 10 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → 𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)})
3130eleq1d 2823 . . . . . . . . 9 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → ( 𝑥𝑆 {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆))
3229, 31imbi12d 345 . . . . . . . 8 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → ((𝑥 ≼ ω → 𝑥𝑆) ↔ ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆)))
3332rspcva 3559 . . . . . . 7 (({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) → ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆))
3423, 28, 33sylsyld 61 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆))
355adantl 482 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴𝑆)
3611adantr 481 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆)
37 ssralv 3987 . . . . . . . 8 (𝐴𝑆 → (∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 → ∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆))
3835, 36, 37sylc 65 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆)
39 dfiun2g 4960 . . . . . . 7 (∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆 𝑧𝐴 ( 𝑆𝑧) = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)})
40 eleq1 2826 . . . . . . 7 ( 𝑧𝐴 ( 𝑆𝑧) = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → ( 𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆 {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆))
4138, 39, 403syl 18 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ( 𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆 {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆))
4234, 41sylibrd 258 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → 𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆))
43 difeq2 4051 . . . . . . 7 (𝑥 = 𝑧𝐴 ( 𝑆𝑧) → ( 𝑆𝑥) = ( 𝑆 𝑧𝐴 ( 𝑆𝑧)))
4443eleq1d 2823 . . . . . 6 (𝑥 = 𝑧𝐴 ( 𝑆𝑧) → (( 𝑆𝑥) ∈ 𝑆 ↔ ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
4544rspccv 3558 . . . . 5 (∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 → ( 𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆 → ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
464, 42, 45sylsyld 61 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
4746adantrd 492 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
4847imp 407 . 2 (((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆)
49 simpr 485 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴 ∈ 𝒫 𝑆)
50 pwuni 4878 . . . . . . 7 𝑆 ⊆ 𝒫 𝑆
515, 50sstrdi 3933 . . . . . 6 (𝐴 ∈ 𝒫 𝑆𝐴 ⊆ 𝒫 𝑆)
52 iundifdifd 30901 . . . . . 6 (𝐴 ⊆ 𝒫 𝑆 → (𝐴 ≠ ∅ → 𝐴 = ( 𝑆 𝑧𝐴 ( 𝑆𝑧))))
5349, 51, 523syl 18 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≠ ∅ → 𝐴 = ( 𝑆 𝑧𝐴 ( 𝑆𝑧))))
5453adantld 491 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → 𝐴 = ( 𝑆 𝑧𝐴 ( 𝑆𝑧))))
55 eleq1 2826 . . . 4 ( 𝐴 = ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) → ( 𝐴𝑆 ↔ ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
5654, 55syl6 35 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ( 𝐴𝑆 ↔ ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆)))
5756imp 407 . 2 (((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ( 𝐴𝑆 ↔ ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
5848, 57mpbird 256 1 (((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → 𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  {cab 2715  wne 2943  wral 3064  wrex 3065  Vcvv 3432  cdif 3884  wss 3887  c0 4256  𝒫 cpw 4533   cuni 4839   cint 4879   ciun 4924   class class class wbr 5074  ran crn 5590  ωcom 7712  cdom 8731  sigAlgebracsiga 32076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-ac2 10219
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-card 9697  df-acn 9700  df-ac 9872  df-siga 32077
This theorem is referenced by:  difelsiga  32101  sigapisys  32123
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