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Theorem sigagenval 34121
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 34120 . . 3 sigaGen = (𝑥 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠})
21a1i 11 . 2 (𝐴𝑉 → sigaGen = (𝑥 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠}))
3 unieq 4923 . . . . . 6 (𝑥 = 𝐴 𝑥 = 𝐴)
43fveq2d 6911 . . . . 5 (𝑥 = 𝐴 → (sigAlgebra‘ 𝑥) = (sigAlgebra‘ 𝐴))
5 sseq1 4021 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑠𝐴𝑠))
64, 5rabeqbidv 3452 . . . 4 (𝑥 = 𝐴 → {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
76inteqd 4956 . . 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 3499 . 2 (𝐴𝑉𝐴 ∈ V)
10 uniexg 7759 . . . . . . 7 (𝐴𝑉 𝐴 ∈ V)
11 pwsiga 34111 . . . . . . 7 ( 𝐴 ∈ V → 𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴))
1210, 11syl 17 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴))
13 pwuni 4950 . . . . . 6 𝐴 ⊆ 𝒫 𝐴
1412, 13jctir 520 . . . . 5 (𝐴𝑉 → (𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ 𝒫 𝐴))
15 sseq2 4022 . . . . . 6 (𝑠 = 𝒫 𝐴 → (𝐴𝑠𝐴 ⊆ 𝒫 𝐴))
1615elrab 3695 . . . . 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 4348 . . 3 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅)
19 intex 5350 . . 3 ({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
2018, 19sylib 218 . 2 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
212, 8, 9, 20fvmptd 7023 1 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  {crab 3433  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912   cint 4951  cmpt 5231  cfv 6563  sigAlgebracsiga 34089  sigaGencsigagen 34119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-siga 34090  df-sigagen 34120
This theorem is referenced by:  sigagensiga  34122  sssigagen  34126  sigagenss  34130
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