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Theorem sigainb 29977
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 4761 . . 3 (𝑆 ran sigAlgebra → (𝑆 ∩ 𝒫 𝐴) ∈ V)
21adantr 481 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ V)
3 inss2 3812 . . 3 (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
43a1i 11 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
5 simpr 477 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴𝑆)
6 pwidg 4144 . . . . 5 (𝐴𝑆𝐴 ∈ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 ∈ 𝒫 𝐴)
85, 7elind 3776 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴))
9 simpll 789 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑆 ran sigAlgebra)
10 simplr 791 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝐴𝑆)
11 inss1 3811 . . . . . . 7 (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆
12 simpr 477 . . . . . . 7 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
1311, 12sseldi 3581 . . . . . 6 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → 𝑥𝑆)
14 difelsiga 29974 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝑥𝑆) → (𝐴𝑥) ∈ 𝑆)
159, 10, 13, 14syl3anc 1323 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ 𝑆)
16 difss 3715 . . . . . . 7 (𝐴𝑥) ⊆ 𝐴
17 elpwg 4138 . . . . . . 7 ((𝐴𝑥) ∈ 𝑆 → ((𝐴𝑥) ∈ 𝒫 𝐴 ↔ (𝐴𝑥) ⊆ 𝐴))
1816, 17mpbiri 248 . . . . . 6 ((𝐴𝑥) ∈ 𝑆 → (𝐴𝑥) ∈ 𝒫 𝐴)
1915, 18syl 17 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ 𝒫 𝐴)
2015, 19elind 3776 . . . 4 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)) → (𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
2120ralrimiva 2960 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → ∀𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴𝑥) ∈ (𝑆 ∩ 𝒫 𝐴))
22 simplll 797 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑆 ran sigAlgebra)
23 simplr 791 . . . . . . . 8 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴))
24 elpwi 4140 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → 𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴))
25 sstr 3591 . . . . . . . . . 10 ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝑆) → 𝑥𝑆)
2611, 25mpan2 706 . . . . . . . . 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 4138 . . . . . . . . 9 (𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑆𝑥𝑆))
2928biimpar 502 . . . . . . . 8 ((𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) ∧ 𝑥𝑆) → 𝑥 ∈ 𝒫 𝑆)
3023, 27, 29syl2anc 692 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝑆)
31 simpr 477 . . . . . . 7 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ≼ ω)
32 sigaclcu 29958 . . . . . . 7 ((𝑆 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆𝑥 ≼ ω) → 𝑥𝑆)
3322, 30, 31, 32syl3anc 1323 . . . . . 6 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥𝑆)
34 sstr 3591 . . . . . . . . 9 ((𝑥 ⊆ (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
353, 34mpan2 706 . . . . . . . 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 4577 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝐴 𝑥𝐴)
38 vuniex 6907 . . . . . . . . 9 𝑥 ∈ V
3938elpw 4136 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴)
4037, 39bitr4i 267 . . . . . . 7 (𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
4136, 40sylib 208 . . . . . 6 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝐴)
4233, 41elind 3776 . . . . 5 ((((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ 𝑥 ≼ ω) → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴))
4342ex 450 . . . 4 (((𝑆 ran sigAlgebra ∧ 𝐴𝑆) ∧ 𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) → (𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
4443ralrimiva 2960 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → ∀𝑥 ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(𝑥 ≼ ω → 𝑥 ∈ (𝑆 ∩ 𝒫 𝐴)))
458, 21, 443jca 1240 . 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 29952 . . 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 502 . 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 1321 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  ran crn 5075  cfv 5847  ωcom 7012  cdom 7897  sigAlgebracsiga 29948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-ac2 9229
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-oi 8359  df-card 8709  df-acn 8712  df-ac 8883  df-cda 8934  df-siga 29949
This theorem is referenced by:  measinb2  30064
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