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Theorem sigaclci 33660
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 33655 . . . . . . . 8 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
21simprd 495 . . . . . . 7 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
32simp2d 1140 . . . . . 6 (𝑆 ran sigAlgebra → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
43adantr 480 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
5 elpwi 4604 . . . . . . . . . . . 12 (𝐴 ∈ 𝒫 𝑆𝐴𝑆)
6 ssrexv 4046 . . . . . . . . . . . 12 (𝐴𝑆 → (∃𝑧𝐴 𝑦 = ( 𝑆𝑧) → ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)))
75, 6syl 17 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 𝑆 → (∃𝑧𝐴 𝑦 = ( 𝑆𝑧) → ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)))
87ss2abdv 4055 . . . . . . . . . 10 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ {𝑦 ∣ ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)})
9 isrnsigau 33655 . . . . . . . . . . . . 13 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → 𝑧𝑆))))
109simprd 495 . . . . . . . . . . . 12 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → 𝑧𝑆)))
1110simp2d 1140 . . . . . . . . . . 11 (𝑆 ran sigAlgebra → ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆)
12 uniiunlem 4079 . . . . . . . . . . . 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 3991 . . . . . . . . 9 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆)
16 abrexexg 7946 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ V)
17 elpwg 4600 . . . . . . . . . . 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 481 . . . . . . . . 9 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆 ↔ {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆))
2015, 19mpbird 257 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆)
212simp3d 1141 . . . . . . . . 9 (𝑆 ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
2221adantr 480 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
2320, 22jca 511 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
24 abrexdom2jm 32254 . . . . . . . . . 10 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ 𝐴)
25 domtr 9005 . . . . . . . . . 10 (({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ 𝐴𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω)
2624, 25sylan 579 . . . . . . . . 9 ((𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω)
2726ex 412 . . . . . . . 8 (𝐴 ∈ 𝒫 𝑆 → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω))
2827adantl 481 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω))
29 breq1 5144 . . . . . . . . 9 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → (𝑥 ≼ ω ↔ {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω))
30 unieq 4913 . . . . . . . . . 10 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → 𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)})
3130eleq1d 2812 . . . . . . . . 9 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → ( 𝑥𝑆 {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆))
3229, 31imbi12d 344 . . . . . . . 8 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → ((𝑥 ≼ ω → 𝑥𝑆) ↔ ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆)))
3332rspcva 3604 . . . . . . 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 481 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴𝑆)
3611adantr 480 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆)
37 ssralv 4045 . . . . . . . 8 (𝐴𝑆 → (∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 → ∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆))
3835, 36, 37sylc 65 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆)
39 dfiun2g 5026 . . . . . . 7 (∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆 𝑧𝐴 ( 𝑆𝑧) = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)})
40 eleq1 2815 . . . . . . 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 259 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → 𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆))
43 difeq2 4111 . . . . . . 7 (𝑥 = 𝑧𝐴 ( 𝑆𝑧) → ( 𝑆𝑥) = ( 𝑆 𝑧𝐴 ( 𝑆𝑧)))
4443eleq1d 2812 . . . . . 6 (𝑥 = 𝑧𝐴 ( 𝑆𝑧) → (( 𝑆𝑥) ∈ 𝑆 ↔ ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
4544rspccv 3603 . . . . 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 491 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
4847imp 406 . 2 (((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆)
49 simpr 484 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴 ∈ 𝒫 𝑆)
50 pwuni 4942 . . . . . . 7 𝑆 ⊆ 𝒫 𝑆
515, 50sstrdi 3989 . . . . . 6 (𝐴 ∈ 𝒫 𝑆𝐴 ⊆ 𝒫 𝑆)
52 iundifdifd 32302 . . . . . 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 490 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → 𝐴 = ( 𝑆 𝑧𝐴 ( 𝑆𝑧))))
55 eleq1 2815 . . . 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 406 . 2 (((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ( 𝐴𝑆 ↔ ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
5848, 57mpbird 257 1 (((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → 𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  {cab 2703  wne 2934  wral 3055  wrex 3064  Vcvv 3468  cdif 3940  wss 3943  c0 4317  𝒫 cpw 4597   cuni 4902   cint 4943   ciun 4990   class class class wbr 5141  ran crn 5670  ωcom 7852  cdom 8939  sigAlgebracsiga 33636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-ac2 10460
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-card 9936  df-acn 9939  df-ac 10113  df-siga 33637
This theorem is referenced by:  difelsiga  33661  sigapisys  33683
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