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Theorem sigainb 33134
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 5320 . . 3 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ (𝑆 ∩ 𝒫 𝐴) ∈ V)
21adantr 482 . 2 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ (𝑆 ∩ 𝒫 𝐴) ∈ V)
3 inss2 4230 . . 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 486 . . . 4 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ 𝐴 ∈ 𝑆)
6 pwidg 4623 . . . . 5 (𝐴 ∈ 𝑆 β†’ 𝐴 ∈ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ 𝐴 ∈ 𝒫 𝐴)
85, 7elind 4195 . . 3 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ 𝐴 ∈ (𝑆 ∩ 𝒫 𝐴))
9 simpll 766 . . . . . 6 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ 𝑆 ∈ βˆͺ ran sigAlgebra)
10 simplr 768 . . . . . 6 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ 𝐴 ∈ 𝑆)
11 inss1 4229 . . . . . . 7 (𝑆 ∩ 𝒫 𝐴) βŠ† 𝑆
12 simpr 486 . . . . . . 7 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴))
1311, 12sselid 3981 . . . . . 6 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ π‘₯ ∈ 𝑆)
14 difelsiga 33131 . . . . . 6 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ π‘₯ ∈ 𝑆) β†’ (𝐴 βˆ– π‘₯) ∈ 𝑆)
159, 10, 13, 14syl3anc 1372 . . . . 5 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ (𝐴 βˆ– π‘₯) ∈ 𝑆)
16 difss 4132 . . . . . . 7 (𝐴 βˆ– π‘₯) βŠ† 𝐴
17 elpwg 4606 . . . . . . 7 ((𝐴 βˆ– π‘₯) ∈ 𝑆 β†’ ((𝐴 βˆ– π‘₯) ∈ 𝒫 𝐴 ↔ (𝐴 βˆ– π‘₯) βŠ† 𝐴))
1816, 17mpbiri 258 . . . . . 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 4195 . . . 4 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ (𝐴 βˆ– π‘₯) ∈ (𝑆 ∩ 𝒫 𝐴))
2120ralrimiva 3147 . . 3 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ βˆ€π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)(𝐴 βˆ– π‘₯) ∈ (𝑆 ∩ 𝒫 𝐴))
22 simplll 774 . . . . . . 7 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ 𝑆 ∈ βˆͺ ran sigAlgebra)
23 simplr 768 . . . . . . . 8 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴))
24 elpwi 4610 . . . . . . . . 9 (π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) β†’ π‘₯ βŠ† (𝑆 ∩ 𝒫 𝐴))
25 sstr 3991 . . . . . . . . . 10 ((π‘₯ βŠ† (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) βŠ† 𝑆) β†’ π‘₯ βŠ† 𝑆)
2611, 25mpan2 690 . . . . . . . . 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 4606 . . . . . . . . 9 (π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) β†’ (π‘₯ ∈ 𝒫 𝑆 ↔ π‘₯ βŠ† 𝑆))
2928biimpar 479 . . . . . . . 8 ((π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) ∧ π‘₯ βŠ† 𝑆) β†’ π‘₯ ∈ 𝒫 𝑆)
3023, 27, 29syl2anc 585 . . . . . . 7 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ π‘₯ ∈ 𝒫 𝑆)
31 simpr 486 . . . . . . 7 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ π‘₯ β‰Ό Ο‰)
32 sigaclcu 33115 . . . . . . 7 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ π‘₯ ∈ 𝒫 𝑆 ∧ π‘₯ β‰Ό Ο‰) β†’ βˆͺ π‘₯ ∈ 𝑆)
3322, 30, 31, 32syl3anc 1372 . . . . . 6 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ βˆͺ π‘₯ ∈ 𝑆)
34 sstr 3991 . . . . . . . . 9 ((π‘₯ βŠ† (𝑆 ∩ 𝒫 𝐴) ∧ (𝑆 ∩ 𝒫 𝐴) βŠ† 𝒫 𝐴) β†’ π‘₯ βŠ† 𝒫 𝐴)
353, 34mpan2 690 . . . . . . . 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 5104 . . . . . . . 8 (π‘₯ βŠ† 𝒫 𝐴 ↔ βˆͺ π‘₯ βŠ† 𝐴)
38 vuniex 7729 . . . . . . . . 9 βˆͺ π‘₯ ∈ V
3938elpw 4607 . . . . . . . 8 (βˆͺ π‘₯ ∈ 𝒫 𝐴 ↔ βˆͺ π‘₯ βŠ† 𝐴)
4037, 39bitr4i 278 . . . . . . 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 4195 . . . . 5 ((((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) ∧ π‘₯ β‰Ό Ο‰) β†’ βˆͺ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴))
4342ex 414 . . . 4 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴)) β†’ (π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)))
4443ralrimiva 3147 . . 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 1129 . 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 33110 . . 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 479 . 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 836 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 397   ∧ w3a 1088   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909   class class class wbr 5149  ran crn 5678  β€˜cfv 6544  Ο‰com 7855   β‰Ό cdom 8937  sigAlgebracsiga 33106
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-ac2 10458
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-dju 9896  df-card 9934  df-acn 9937  df-ac 10111  df-siga 33107
This theorem is referenced by:  measinb2  33221
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