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Theorem sigainb 33599
Description: Building a sigma-algebra from intersections with a given set. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
sigainb ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))

Proof of Theorem sigainb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 inex1g 5319 . . 3 (𝑆 ran sigAlgebra → (𝑆 ∩ 𝒫 𝐴) ∈ V)
21adantr 480 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ V)
3 inss2 4229 . . 3 (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
43a1i 11 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
5 simpr 484 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴𝑆)
6 pwidg 4622 . . . . 5 (𝐴𝑆𝐴 ∈ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 ∈ 𝒫 𝐴)
85, 7elind 4194 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴))
9 simpll 764 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑆 ran sigAlgebra)
10 simplr 766 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝐴𝑆)
11 inss1 4228 . . . . . . 7 (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆
12 simpr 484 . . . . . . 7 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
1311, 12sselid 3980 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥𝑆)
14 difelsiga 33596 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝑥𝑆) → (𝐴𝑥) ∈ 𝑆)
159, 10, 13, 14syl3anc 1370 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ 𝑆)
16 difss 4131 . . . . . . 7 (𝐴𝑥) ⊆ 𝐴
17 elpwg 4605 . . . . . . 7 ((𝐴𝑥) ∈ 𝑆 → ((𝐴𝑥) ∈ 𝒫 𝐴 ↔ (𝐴𝑥) ⊆ 𝐴))
1816, 17mpbiri 258 . . . . . 6 ((𝐴𝑥) ∈ 𝑆 → (𝐴𝑥) ∈ 𝒫 𝐴)
1915, 18syl 17 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ 𝒫 𝐴)
2015, 19elind 4194 . . . 4 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
2120ralrimiva 3145 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
22 simplll 772 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑆 ran sigAlgebra)
23 simplr 766 . . . . . . . 8 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴))
24 elpwi 4609 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴))
25 sstr 3990 . . . . . . . . . 10 ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆) → 𝑥𝑆)
2611, 25mpan2 688 . . . . . . . . 9 (𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) → 𝑥𝑆)
2723, 24, 263syl 18 . . . . . . . 8 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥𝑆)
28 elpwg 4605 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑆𝑥𝑆))
2928biimpar 477 . . . . . . . 8 ((𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) ∧ 𝑥𝑆) → 𝑥 ∈ 𝒫 𝑆)
3023, 27, 29syl2anc 583 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝑆)
31 simpr 484 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ≼ ω)
32 sigaclcu 33580 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆𝑥 ≼ ω) → 𝑥𝑆)
3322, 30, 31, 32syl3anc 1370 . . . . . 6 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥𝑆)
34 sstr 3990 . . . . . . . . 9 ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
353, 34mpan2 688 . . . . . . . 8 (𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
3623, 24, 353syl 18 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ⊆ 𝒫 𝐴)
37 sspwuni 5103 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝐴 𝑥𝐴)
38 vuniex 7733 . . . . . . . . 9 𝑥 ∈ V
3938elpw 4606 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴)
4037, 39bitr4i 278 . . . . . . 7 (𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
4136, 40sylib 217 . . . . . 6 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝐴)
4233, 41elind 4194 . . . . 5 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
4342ex 412 . . . 4 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) → (𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
4443ralrimiva 3145 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
458, 21, 443jca 1127 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))))
46 issiga 33575 . . 3 ((𝑆 ∩ 𝒫 𝐴) ∈ V → ((𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴) ↔ ((𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))))))
4746biimpar 477 . 2 (((𝑆 ∩ 𝒫 𝐴) ∈ V ∧ ((𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))))) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))
482, 4, 45, 47syl12anc 834 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wcel 2105  wral 3060  Vcvv 3473  cdif 3945  cin 3947  wss 3948  𝒫 cpw 4602   cuni 4908   class class class wbr 5148  ran crn 5677  cfv 6543  ωcom 7859  cdom 8943  sigAlgebracsiga 33571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-ac2 10464
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-oi 9511  df-dju 9902  df-card 9940  df-acn 9943  df-ac 10117  df-siga 33572
This theorem is referenced by:  measinb2  33686
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