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Theorem sigaclci 31620
 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 31615 . . . . . . . 8 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
21simprd 500 . . . . . . 7 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
32simp2d 1141 . . . . . 6 (𝑆 ran sigAlgebra → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
43adantr 485 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
5 elpwi 4504 . . . . . . . . . . . 12 (𝐴 ∈ 𝒫 𝑆𝐴𝑆)
6 ssrexv 3960 . . . . . . . . . . . 12 (𝐴𝑆 → (∃𝑧𝐴 𝑦 = ( 𝑆𝑧) → ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)))
75, 6syl 17 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 𝑆 → (∃𝑧𝐴 𝑦 = ( 𝑆𝑧) → ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)))
87ss2abdv 3969 . . . . . . . . . 10 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ {𝑦 ∣ ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)})
9 isrnsigau 31615 . . . . . . . . . . . . 13 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → 𝑧𝑆))))
109simprd 500 . . . . . . . . . . . 12 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → 𝑧𝑆)))
1110simp2d 1141 . . . . . . . . . . 11 (𝑆 ran sigAlgebra → ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆)
12 uniiunlem 3991 . . . . . . . . . . . 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 235 . . . . . . . . . 10 (𝑆 ran sigAlgebra → {𝑦 ∣ ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆)
158, 14sylan9ssr 3907 . . . . . . . . 9 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆)
16 abrexexg 7667 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ V)
17 elpwg 4498 . . . . . . . . . . 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 486 . . . . . . . . 9 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆 ↔ {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆))
2015, 19mpbird 260 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆)
212simp3d 1142 . . . . . . . . 9 (𝑆 ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
2221adantr 485 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
2320, 22jca 516 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝒫 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
24 abrexdom2jm 30376 . . . . . . . . . 10 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ 𝐴)
25 domtr 8581 . . . . . . . . . 10 (({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ 𝐴𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω)
2624, 25sylan 584 . . . . . . . . 9 ((𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω)
2726ex 417 . . . . . . . 8 (𝐴 ∈ 𝒫 𝑆 → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω))
2827adantl 486 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω))
29 breq1 5036 . . . . . . . . 9 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → (𝑥 ≼ ω ↔ {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω))
30 unieq 4810 . . . . . . . . . 10 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → 𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)})
3130eleq1d 2837 . . . . . . . . 9 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → ( 𝑥𝑆 {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆))
3229, 31imbi12d 349 . . . . . . . 8 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → ((𝑥 ≼ ω → 𝑥𝑆) ↔ ({𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ 𝑆)))
3332rspcva 3540 . . . . . . 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 486 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴𝑆)
3611adantr 485 . . . . . . . 8 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆)
37 ssralv 3959 . . . . . . . 8 (𝐴𝑆 → (∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 → ∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆))
3835, 36, 37sylc 65 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆)
39 dfiun2g 4920 . . . . . . 7 (∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆 𝑧𝐴 ( 𝑆𝑧) = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)})
40 eleq1 2840 . . . . . . 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 262 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → 𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆))
43 difeq2 4023 . . . . . . 7 (𝑥 = 𝑧𝐴 ( 𝑆𝑧) → ( 𝑆𝑥) = ( 𝑆 𝑧𝐴 ( 𝑆𝑧)))
4443eleq1d 2837 . . . . . 6 (𝑥 = 𝑧𝐴 ( 𝑆𝑧) → (( 𝑆𝑥) ∈ 𝑆 ↔ ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
4544rspccv 3539 . . . . 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 496 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
4847imp 411 . 2 (((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆)
49 simpr 489 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴 ∈ 𝒫 𝑆)
50 pwuni 4838 . . . . . . 7 𝑆 ⊆ 𝒫 𝑆
515, 50sstrdi 3905 . . . . . 6 (𝐴 ∈ 𝒫 𝑆𝐴 ⊆ 𝒫 𝑆)
52 iundifdifd 30424 . . . . . 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 495 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → 𝐴 = ( 𝑆 𝑧𝐴 ( 𝑆𝑧))))
55 eleq1 2840 . . . 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 411 . 2 (((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ( 𝐴𝑆 ↔ ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
5848, 57mpbird 260 1 (((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → 𝐴𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  {cab 2736   ≠ wne 2952  ∀wral 3071  ∃wrex 3072  Vcvv 3410   ∖ cdif 3856   ⊆ wss 3859  ∅c0 4226  𝒫 cpw 4495  ∪ cuni 4799  ∩ cint 4839  ∪ ciun 4884   class class class wbr 5033  ran crn 5526  ωcom 7580   ≼ cdom 8526  sigAlgebracsiga 31596 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-ac2 9924 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-se 5485  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-isom 6345  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-er 8300  df-map 8419  df-en 8529  df-dom 8530  df-card 9402  df-acn 9405  df-ac 9577  df-siga 31597 This theorem is referenced by:  difelsiga  31621  sigapisys  31643
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