Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sigagenval Structured version   Visualization version   GIF version

Theorem sigagenval 33668
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 33667 . . 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 4913 . . . . . 6 (π‘₯ = 𝐴 β†’ βˆͺ π‘₯ = βˆͺ 𝐴)
43fveq2d 6889 . . . . 5 (π‘₯ = 𝐴 β†’ (sigAlgebraβ€˜βˆͺ π‘₯) = (sigAlgebraβ€˜βˆͺ 𝐴))
5 sseq1 4002 . . . . 5 (π‘₯ = 𝐴 β†’ (π‘₯ βŠ† 𝑠 ↔ 𝐴 βŠ† 𝑠))
64, 5rabeqbidv 3443 . . . 4 (π‘₯ = 𝐴 β†’ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ π‘₯) ∣ π‘₯ βŠ† 𝑠} = {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
76inteqd 4948 . . 3 (π‘₯ = 𝐴 β†’ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ π‘₯) ∣ π‘₯ βŠ† 𝑠} = ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
87adantl 481 . 2 ((𝐴 ∈ 𝑉 ∧ π‘₯ = 𝐴) β†’ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ π‘₯) ∣ π‘₯ βŠ† 𝑠} = ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
9 elex 3487 . 2 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ V)
10 uniexg 7727 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ βˆͺ 𝐴 ∈ V)
11 pwsiga 33658 . . . . . . 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 4942 . . . . . 6 𝐴 βŠ† 𝒫 βˆͺ 𝐴
1412, 13jctir 520 . . . . 5 (𝐴 ∈ 𝑉 β†’ (𝒫 βˆͺ 𝐴 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∧ 𝐴 βŠ† 𝒫 βˆͺ 𝐴))
15 sseq2 4003 . . . . . 6 (𝑠 = 𝒫 βˆͺ 𝐴 β†’ (𝐴 βŠ† 𝑠 ↔ 𝐴 βŠ† 𝒫 βˆͺ 𝐴))
1615elrab 3678 . . . . 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 4330 . . 3 (𝐴 ∈ 𝑉 β†’ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠} β‰  βˆ…)
19 intex 5330 . . 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 6999 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 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  {crab 3426  Vcvv 3468   βŠ† wss 3943  βˆ…c0 4317  π’« cpw 4597  βˆͺ cuni 4902  βˆ© cint 4943   ↦ cmpt 5224  β€˜cfv 6537  sigAlgebracsiga 33636  sigaGencsigagen 33666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-siga 33637  df-sigagen 33667
This theorem is referenced by:  sigagensiga  33669  sssigagen  33673  sigagenss  33677
  Copyright terms: Public domain W3C validator