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Theorem sigagenval 31077
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 31076 . . 3 sigaGen = (𝑥 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠})
21a1i 11 . 2 (𝐴𝑉 → sigaGen = (𝑥 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠}))
3 unieq 4717 . . . . . 6 (𝑥 = 𝐴 𝑥 = 𝐴)
43fveq2d 6501 . . . . 5 (𝑥 = 𝐴 → (sigAlgebra‘ 𝑥) = (sigAlgebra‘ 𝐴))
5 sseq1 3877 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑠𝐴𝑠))
64, 5rabeqbidv 3403 . . . 4 (𝑥 = 𝐴 → {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
76inteqd 4751 . . 3 (𝑥 = 𝐴 {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
87adantl 474 . 2 ((𝐴𝑉𝑥 = 𝐴) → {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
9 elex 3428 . 2 (𝐴𝑉𝐴 ∈ V)
10 uniexg 7284 . . . . . . 7 (𝐴𝑉 𝐴 ∈ V)
11 pwsiga 31067 . . . . . . 7 ( 𝐴 ∈ V → 𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴))
1210, 11syl 17 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴))
13 pwuni 4745 . . . . . 6 𝐴 ⊆ 𝒫 𝐴
1412, 13jctir 513 . . . . 5 (𝐴𝑉 → (𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ 𝒫 𝐴))
15 sseq2 3878 . . . . . 6 (𝑠 = 𝒫 𝐴 → (𝐴𝑠𝐴 ⊆ 𝒫 𝐴))
1615elrab 3590 . . . . 5 (𝒫 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ↔ (𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ 𝒫 𝐴))
1714, 16sylibr 226 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
1817ne0d 4182 . . 3 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅)
19 intex 5093 . . 3 ({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
2018, 19sylib 210 . 2 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
212, 8, 9, 20fvmptd 6600 1 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wcel 2051  wne 2962  {crab 3087  Vcvv 3410  wss 3824  c0 4173  𝒫 cpw 4417   cuni 4709   cint 4746  cmpt 5005  cfv 6186  sigAlgebracsiga 31044  sigaGencsigagen 31075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-int 4747  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-iota 6150  df-fun 6188  df-fv 6194  df-siga 31045  df-sigagen 31076
This theorem is referenced by:  sigagensiga  31078  sssigagen  31082  sigagenss  31086
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