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Theorem sigainb 33203
Description: Building a sigma-algebra from intersections with a given set. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
sigainb ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebraβ€˜π΄))
Proof of Theorem sigainb
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 inex1g 5319 . . 3 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ (𝑆 ∩ 𝒫 𝐴) ∈ V)
21adantr 481 . 2 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ (𝑆 ∩ 𝒫 𝐴) ∈ V)
3 inss2 4229 . . 3 (𝑆 ∩ 𝒫 𝐴) βŠ† 𝒫 𝐴
43a1i 11 . 2 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ (𝑆 ∩ 𝒫 𝐴) βŠ† 𝒫 𝐴)
5 simpr 485 . . . 4 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ 𝐴 ∈ 𝑆)
6 pwidg 4622 . . . . 5 (𝐴 ∈ 𝑆 β†’ 𝐴 ∈ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ 𝐴 ∈ 𝒫 𝐴)
85, 7elind 4194 . . 3 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴))
9 simpll 765 . . . . . 6 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ 𝑆 ∈ βˆͺ ran sigAlgebra)
10 simplr 767 . . . . . 6 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ 𝐴 ∈ 𝑆)
11 inss1 4228 . . . . . . 7 (𝑆 ∩ 𝒫 𝐴) βŠ† 𝑆
12 simpr 485 . . . . . . 7 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴))
1311, 12sselid 3980 . . . . . 6 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ π‘₯ ∈ 𝑆)
14 difelsiga 33200 . . . . . 6 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ π‘₯ ∈ 𝑆) β†’ (𝐴 βˆ– π‘₯) ∈ 𝑆)
159, 10, 13, 14syl3anc 1371 . . . . 5 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ (𝐴 βˆ– π‘₯) ∈ 𝑆)
16 difss 4131 . . . . . . 7 (𝐴 βˆ– π‘₯) βŠ† 𝐴
17 elpwg 4605 . . . . . . 7 ((𝐴 βˆ– π‘₯) ∈ 𝑆 β†’ ((𝐴 βˆ– π‘₯) ∈ 𝒫 𝐴 ↔ (𝐴 βˆ– π‘₯) βŠ† 𝐴))
1816, 17mpbiri 257 . . . . . 6 ((𝐴 βˆ– π‘₯) ∈ 𝑆 β†’ (𝐴 βˆ– π‘₯) ∈ 𝒫 𝐴)
1915, 18syl 17 . . . . 5 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ (𝐴 βˆ– π‘₯) ∈ 𝒫 𝐴)
2015, 19elind 4194 . . . 4 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ (𝐴 βˆ– π‘₯) ∈ (𝑆 ∩ 𝒫 𝐴))
2120ralrimiva 3146 . . 3 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ βˆ€π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 βˆ– π‘₯) ∈ (𝑆 ∩ 𝒫 𝐴))
22 simplll 773 . . . . . . 7 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ 𝑆 ∈ βˆͺ ran sigAlgebra)
23 simplr 767 . . . . . . . 8 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴))
24 elpwi 4609 . . . . . . . . 9 (π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) β†’ π‘₯ βŠ† (𝑆 ∩ 𝒫 𝐴))
25 sstr 3990 . . . . . . . . . 10 ((π‘₯ βŠ† (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) βŠ† 𝑆) β†’ π‘₯ βŠ† 𝑆)
2611, 25mpan2 689 . . . . . . . . 9 (π‘₯ βŠ† (𝑆 ∩ 𝒫 𝐴) β†’ π‘₯ βŠ† 𝑆)
2723, 24, 263syl 18 . . . . . . . 8 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ π‘₯ βŠ† 𝑆)
28 elpwg 4605 . . . . . . . . 9 (π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) β†’ (π‘₯ ∈ 𝒫 𝑆 ↔ π‘₯ βŠ† 𝑆))
2928biimpar 478 . . . . . . . 8 ((π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) ∧ π‘₯ βŠ† 𝑆) β†’ π‘₯ ∈ 𝒫 𝑆)
3023, 27, 29syl2anc 584 . . . . . . 7 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ π‘₯ ∈ 𝒫 𝑆)
31 simpr 485 . . . . . . 7 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ π‘₯ β‰Ό Ο‰)
32 sigaclcu 33184 . . . . . . 7 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ π‘₯ ∈ 𝒫 𝑆 ∧ π‘₯ β‰Ό Ο‰) β†’ βˆͺ π‘₯ ∈ 𝑆)
3322, 30, 31, 32syl3anc 1371 . . . . . 6 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ βˆͺ π‘₯ ∈ 𝑆)
34 sstr 3990 . . . . . . . . 9 ((π‘₯ βŠ† (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) βŠ† 𝒫 𝐴) β†’ π‘₯ βŠ† 𝒫 𝐴)
353, 34mpan2 689 . . . . . . . 8 (π‘₯ βŠ† (𝑆 ∩ 𝒫 𝐴) β†’ π‘₯ βŠ† 𝒫 𝐴)
3623, 24, 353syl 18 . . . . . . 7 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ π‘₯ βŠ† 𝒫 𝐴)
37 sspwuni 5103 . . . . . . . 8 (π‘₯ βŠ† 𝒫 𝐴 ↔ βˆͺ π‘₯ βŠ† 𝐴)
38 vuniex 7731 . . . . . . . . 9 βˆͺ π‘₯ ∈ V
3938elpw 4606 . . . . . . . 8 (βˆͺ π‘₯ ∈ 𝒫 𝐴 ↔ βˆͺ π‘₯ βŠ† 𝐴)
4037, 39bitr4i 277 . . . . . . 7 (π‘₯ βŠ† 𝒫 𝐴 ↔ βˆͺ π‘₯ ∈ 𝒫 𝐴)
4136, 40sylib 217 . . . . . 6 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ βˆͺ π‘₯ ∈ 𝒫 𝐴)
4233, 41elind 4194 . . . . 5 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ βˆͺ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴))
4342ex 413 . . . 4 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) β†’ (π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)))
4443ralrimiva 3146 . . 3 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ βˆ€π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)))
458, 21, 443jca 1128 . 2 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ βˆ€π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 βˆ– π‘₯) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ βˆ€π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴))))
46 issiga 33179 . . 3 ((𝑆 ∩ 𝒫 𝐴) ∈ V β†’ ((𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebraβ€˜π΄) ↔ ((𝑆 ∩ 𝒫 𝐴) βŠ† 𝒫 𝐴 ∧ (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ βˆ€π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 βˆ– π‘₯) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ βˆ€π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴))))))
4746biimpar 478 . 2 (((𝑆 ∩ 𝒫 𝐴) ∈ V ∧ ((𝑆 ∩ 𝒫 𝐴) βŠ† 𝒫 𝐴 ∧ (𝐴 ∈ (𝑆 ∩ 𝒫 𝐴) ∧ βˆ€π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 βˆ– π‘₯) ∈ (𝑆 ∩ 𝒫 𝐴) ∧ βˆ€π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴))))) β†’ (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebraβ€˜π΄))
482, 4, 45, 47syl12anc 835 1 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebraβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βˆ– cdif 3945   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908   class class class wbr 5148  ran crn 5677  β€˜cfv 6543  Ο‰com 7857   β‰Ό cdom 8939  sigAlgebracsiga 33175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-ac2 10460
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-dju 9898  df-card 9936  df-acn 9939  df-ac 10113  df-siga 33176
This theorem is referenced by:  measinb2  33290
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