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Theorem sigaclci 31290
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 31285 . . . . . . . 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 1135 . . . . . 6 (𝑆 ran sigAlgebra → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
43adantr 481 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
5 elpwi 4547 . . . . . . . . . . . 12 (𝐴 ∈ 𝒫 𝑆𝐴𝑆)
6 ssrexv 4031 . . . . . . . . . . . 12 (𝐴𝑆 → (∃𝑧𝐴 𝑦 = ( 𝑆𝑧) → ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)))
75, 6syl 17 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 𝑆 → (∃𝑧𝐴 𝑦 = ( 𝑆𝑧) → ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)))
87ss2abdv 4041 . . . . . . . . . 10 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ {𝑦 ∣ ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)})
9 isrnsigau 31285 . . . . . . . . . . . . 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 1135 . . . . . . . . . . 11 (𝑆 ran sigAlgebra → ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆)
12 uniiunlem 4058 . . . . . . . . . . . 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 233 . . . . . . . . . 10 (𝑆 ran sigAlgebra → {𝑦 ∣ ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆)
158, 14sylan9ssr 3978 . . . . . . . . 9 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆)
16 abrexexg 7651 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ V)
17 elpwg 4541 . . . . . . . . . . 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 258 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆)
212simp3d 1136 . . . . . . . . 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 30195 . . . . . . . . . 10 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ 𝐴)
25 domtr 8550 . . . . . . . . . 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 5060 . . . . . . . . 9 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → (𝑥 ≼ ω ↔ {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω))
30 unieq 4838 . . . . . . . . . 10 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → 𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)})
3130eleq1d 2894 . . . . . . . . 9 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → ( 𝑥𝑆 {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆))
3229, 31imbi12d 346 . . . . . . . 8 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → ((𝑥 ≼ ω → 𝑥𝑆) ↔ ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆)))
3332rspcva 3618 . . . . . . 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 4030 . . . . . . . 8 (𝐴𝑆 → (∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 → ∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆))
3835, 36, 37sylc 65 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆)
39 dfiun2g 4946 . . . . . . 7 (∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆 𝑧𝐴 ( 𝑆𝑧) = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)})
40 eleq1 2897 . . . . . . 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 260 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → 𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆))
43 difeq2 4090 . . . . . . 7 (𝑥 = 𝑧𝐴 ( 𝑆𝑧) → ( 𝑆𝑥) = ( 𝑆 𝑧𝐴 ( 𝑆𝑧)))
4443eleq1d 2894 . . . . . 6 (𝑥 = 𝑧𝐴 ( 𝑆𝑧) → (( 𝑆𝑥) ∈ 𝑆 ↔ ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
4544rspccv 3617 . . . . 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 4866 . . . . . . 7 𝑆 ⊆ 𝒫 𝑆
515, 50sstrdi 3976 . . . . . 6 (𝐴 ∈ 𝒫 𝑆𝐴 ⊆ 𝒫 𝑆)
52 iundifdifd 30241 . . . . . 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 2897 . . . 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 258 1 (((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → 𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wral 3135  wrex 3136  Vcvv 3492  cdif 3930  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830   cint 4867   ciun 4910   class class class wbr 5057  ran crn 5549  ωcom 7569  cdom 8495  sigAlgebracsiga 31266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-ac2 9873
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-card 9356  df-acn 9359  df-ac 9530  df-siga 31267
This theorem is referenced by:  difelsiga  31291  sigapisys  31313
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