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Theorem sigagenval 34142
Description: Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
sigagenval (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
Distinct variable group:   𝐴,𝑠
Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem sigagenval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-sigagen 34141 . . 3 sigaGen = (𝑥 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠})
21a1i 11 . 2 (𝐴𝑉 → sigaGen = (𝑥 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠}))
3 unieq 4917 . . . . . 6 (𝑥 = 𝐴 𝑥 = 𝐴)
43fveq2d 6909 . . . . 5 (𝑥 = 𝐴 → (sigAlgebra‘ 𝑥) = (sigAlgebra‘ 𝐴))
5 sseq1 4008 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑠𝐴𝑠))
64, 5rabeqbidv 3454 . . . 4 (𝑥 = 𝐴 → {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
76inteqd 4950 . . 3 (𝑥 = 𝐴 {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
87adantl 481 . 2 ((𝐴𝑉𝑥 = 𝐴) → {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
9 elex 3500 . 2 (𝐴𝑉𝐴 ∈ V)
10 uniexg 7761 . . . . . . 7 (𝐴𝑉 𝐴 ∈ V)
11 pwsiga 34132 . . . . . . 7 ( 𝐴 ∈ V → 𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴))
1210, 11syl 17 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴))
13 pwuni 4944 . . . . . 6 𝐴 ⊆ 𝒫 𝐴
1412, 13jctir 520 . . . . 5 (𝐴𝑉 → (𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ 𝒫 𝐴))
15 sseq2 4009 . . . . . 6 (𝑠 = 𝒫 𝐴 → (𝐴𝑠𝐴 ⊆ 𝒫 𝐴))
1615elrab 3691 . . . . 5 (𝒫 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ↔ (𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ 𝒫 𝐴))
1714, 16sylibr 234 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
1817ne0d 4341 . . 3 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅)
19 intex 5343 . . 3 ({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
2018, 19sylib 218 . 2 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
212, 8, 9, 20fvmptd 7022 1 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wne 2939  {crab 3435  Vcvv 3479  wss 3950  c0 4332  𝒫 cpw 4599   cuni 4906   cint 4945  cmpt 5224  cfv 6560  sigAlgebracsiga 34110  sigaGencsigagen 34140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-siga 34111  df-sigagen 34141
This theorem is referenced by:  sigagensiga  34143  sssigagen  34147  sigagenss  34151
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