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Theorem sigaclci 31008
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 31003 . . . . . . . 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 1136 . . . . . 6 (𝑆 ran sigAlgebra → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
43adantr 481 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆)
5 elpwi 4463 . . . . . . . . . . . 12 (𝐴 ∈ 𝒫 𝑆𝐴𝑆)
6 ssrexv 3955 . . . . . . . . . . . 12 (𝐴𝑆 → (∃𝑧𝐴 𝑦 = ( 𝑆𝑧) → ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)))
75, 6syl 17 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 𝑆 → (∃𝑧𝐴 𝑦 = ( 𝑆𝑧) → ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)))
87ss2abdv 3965 . . . . . . . . . 10 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ {𝑦 ∣ ∃𝑧𝑆 𝑦 = ( 𝑆𝑧)})
9 isrnsigau 31003 . . . . . . . . . . . . 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 1136 . . . . . . . . . . 11 (𝑆 ran sigAlgebra → ∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆)
12 uniiunlem 3982 . . . . . . . . . . . 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 3903 . . . . . . . . 9 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ⊆ 𝑆)
16 abrexexg 7518 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ∈ V)
17 elpwg 4461 . . . . . . . . . . 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 1137 . . . . . . . . 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 29960 . . . . . . . . . 10 (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ 𝐴)
25 domtr 8410 . . . . . . . . . 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 4965 . . . . . . . . 9 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → (𝑥 ≼ ω ↔ {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} ≼ ω))
30 unieq 4753 . . . . . . . . . 10 (𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)} → 𝑥 = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)})
3130eleq1d 2867 . . . . . . . . 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 3557 . . . . . . 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 3954 . . . . . . . 8 (𝐴𝑆 → (∀𝑧𝑆 ( 𝑆𝑧) ∈ 𝑆 → ∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆))
3835, 36, 37sylc 65 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆)
39 dfiun2g 4858 . . . . . . 7 (∀𝑧𝐴 ( 𝑆𝑧) ∈ 𝑆 𝑧𝐴 ( 𝑆𝑧) = {𝑦 ∣ ∃𝑧𝐴 𝑦 = ( 𝑆𝑧)})
40 eleq1 2870 . . . . . . 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 4014 . . . . . . 7 (𝑥 = 𝑧𝐴 ( 𝑆𝑧) → ( 𝑆𝑥) = ( 𝑆 𝑧𝐴 ( 𝑆𝑧)))
4443eleq1d 2867 . . . . . 6 (𝑥 = 𝑧𝐴 ( 𝑆𝑧) → (( 𝑆𝑥) ∈ 𝑆 ↔ ( 𝑆 𝑧𝐴 ( 𝑆𝑧)) ∈ 𝑆))
4544rspccv 3556 . . . . 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 4781 . . . . . . 7 𝑆 ⊆ 𝒫 𝑆
515, 50syl6ss 3901 . . . . . 6 (𝐴 ∈ 𝒫 𝑆𝐴 ⊆ 𝒫 𝑆)
52 iundifdifd 30003 . . . . . 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 2870 . . . 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 1080   = wceq 1522  wcel 2081  {cab 2775  wne 2984  wral 3105  wrex 3106  Vcvv 3437  cdif 3856  wss 3859  c0 4211  𝒫 cpw 4453   cuni 4745   cint 4782   ciun 4825   class class class wbr 4962  ran crn 5444  ωcom 7436  cdom 8355  sigAlgebracsiga 30984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-ac2 9731
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-card 9214  df-acn 9217  df-ac 9388  df-siga 30985
This theorem is referenced by:  difelsiga  31009  sigapisys  31031
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