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Theorem sigagenval 32779
Description: Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
sigagenval (𝐴 ∈ 𝑉 β†’ (sigaGenβ€˜π΄) = ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
Distinct variable group:   𝐴,𝑠
Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem sigagenval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 df-sigagen 32778 . . 3 sigaGen = (π‘₯ ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ π‘₯) ∣ π‘₯ βŠ† 𝑠})
21a1i 11 . 2 (𝐴 ∈ 𝑉 β†’ sigaGen = (π‘₯ ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ π‘₯) ∣ π‘₯ βŠ† 𝑠}))
3 unieq 4881 . . . . . 6 (π‘₯ = 𝐴 β†’ βˆͺ π‘₯ = βˆͺ 𝐴)
43fveq2d 6851 . . . . 5 (π‘₯ = 𝐴 β†’ (sigAlgebraβ€˜βˆͺ π‘₯) = (sigAlgebraβ€˜βˆͺ 𝐴))
5 sseq1 3974 . . . . 5 (π‘₯ = 𝐴 β†’ (π‘₯ βŠ† 𝑠 ↔ 𝐴 βŠ† 𝑠))
64, 5rabeqbidv 3427 . . . 4 (π‘₯ = 𝐴 β†’ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ π‘₯) ∣ π‘₯ βŠ† 𝑠} = {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
76inteqd 4917 . . 3 (π‘₯ = 𝐴 β†’ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ π‘₯) ∣ π‘₯ βŠ† 𝑠} = ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
87adantl 483 . 2 ((𝐴 ∈ 𝑉 ∧ π‘₯ = 𝐴) β†’ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ π‘₯) ∣ π‘₯ βŠ† 𝑠} = ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
9 elex 3466 . 2 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ V)
10 uniexg 7682 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ βˆͺ 𝐴 ∈ V)
11 pwsiga 32769 . . . . . . 7 (βˆͺ 𝐴 ∈ V β†’ 𝒫 βˆͺ 𝐴 ∈ (sigAlgebraβ€˜βˆͺ 𝐴))
1210, 11syl 17 . . . . . 6 (𝐴 ∈ 𝑉 β†’ 𝒫 βˆͺ 𝐴 ∈ (sigAlgebraβ€˜βˆͺ 𝐴))
13 pwuni 4911 . . . . . 6 𝐴 βŠ† 𝒫 βˆͺ 𝐴
1412, 13jctir 522 . . . . 5 (𝐴 ∈ 𝑉 β†’ (𝒫 βˆͺ 𝐴 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∧ 𝐴 βŠ† 𝒫 βˆͺ 𝐴))
15 sseq2 3975 . . . . . 6 (𝑠 = 𝒫 βˆͺ 𝐴 β†’ (𝐴 βŠ† 𝑠 ↔ 𝐴 βŠ† 𝒫 βˆͺ 𝐴))
1615elrab 3650 . . . . 5 (𝒫 βˆͺ 𝐴 ∈ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠} ↔ (𝒫 βˆͺ 𝐴 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∧ 𝐴 βŠ† 𝒫 βˆͺ 𝐴))
1714, 16sylibr 233 . . . 4 (𝐴 ∈ 𝑉 β†’ 𝒫 βˆͺ 𝐴 ∈ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
1817ne0d 4300 . . 3 (𝐴 ∈ 𝑉 β†’ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠} β‰  βˆ…)
19 intex 5299 . . 3 ({𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠} β‰  βˆ… ↔ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠} ∈ V)
2018, 19sylib 217 . 2 (𝐴 ∈ 𝑉 β†’ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠} ∈ V)
212, 8, 9, 20fvmptd 6960 1 (𝐴 ∈ 𝑉 β†’ (sigaGenβ€˜π΄) = ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  {crab 3410  Vcvv 3448   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565  βˆͺ cuni 4870  βˆ© cint 4912   ↦ cmpt 5193  β€˜cfv 6501  sigAlgebracsiga 32747  sigaGencsigagen 32777
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-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  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-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-siga 32748  df-sigagen 32778
This theorem is referenced by:  sigagensiga  32780  sssigagen  32784  sigagenss  32788
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