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Theorem sigainb 32775
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 5281 . . 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 4194 . . 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 4585 . . . . 5 (𝐴 ∈ 𝑆 β†’ 𝐴 ∈ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) β†’ 𝐴 ∈ 𝒫 𝐴)
85, 7elind 4159 . . 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 4193 . . . . . . 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 3947 . . . . . 6 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ π‘₯ ∈ 𝑆)
14 difelsiga 32772 . . . . . 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 4096 . . . . . . 7 (𝐴 βˆ– π‘₯) βŠ† 𝐴
17 elpwg 4568 . . . . . . 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 4159 . . . 4 (((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) ∧ π‘₯ ∈ (𝑆 ∩ 𝒫 𝐴)) β†’ (𝐴 βˆ– π‘₯) ∈ (𝑆 ∩ 𝒫 𝐴))
2120ralrimiva 3144 . . 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 4572 . . . . . . . . 9 (π‘₯ ∈ 𝒫 (𝑆 ∩ 𝒫 𝐴) β†’ π‘₯ βŠ† (𝑆 ∩ 𝒫 𝐴))
25 sstr 3957 . . . . . . . . . 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 4568 . . . . . . . . 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 32756 . . . . . . 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 3957 . . . . . . . . 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 5065 . . . . . . . 8 (π‘₯ βŠ† 𝒫 𝐴 ↔ βˆͺ π‘₯ βŠ† 𝐴)
38 vuniex 7681 . . . . . . . . 9 βˆͺ π‘₯ ∈ V
3938elpw 4569 . . . . . . . 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 4159 . . . . 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 3144 . . 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 32751 . . 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 3065  Vcvv 3448   βˆ– cdif 3912   ∩ cin 3914   βŠ† wss 3915  π’« cpw 4565  βˆͺ cuni 4870   class class class wbr 5110  ran crn 5639  β€˜cfv 6501  Ο‰com 7807   β‰Ό cdom 8888  sigAlgebracsiga 32747
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-ac2 10406
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9453  df-dju 9844  df-card 9882  df-acn 9885  df-ac 10059  df-siga 32748
This theorem is referenced by:  measinb2  32862
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