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Theorem sigagenval 30008
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 30007 . . 3 sigaGen = (𝑥 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠})
21a1i 11 . 2 (𝐴𝑉 → sigaGen = (𝑥 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠}))
3 unieq 4415 . . . . . 6 (𝑥 = 𝐴 𝑥 = 𝐴)
43fveq2d 6157 . . . . 5 (𝑥 = 𝐴 → (sigAlgebra‘ 𝑥) = (sigAlgebra‘ 𝐴))
5 sseq1 3610 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑠𝐴𝑠))
64, 5rabeqbidv 3184 . . . 4 (𝑥 = 𝐴 → {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
76inteqd 4450 . . 3 (𝑥 = 𝐴 {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
87adantl 482 . 2 ((𝐴𝑉𝑥 = 𝐴) → {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
9 elex 3201 . 2 (𝐴𝑉𝐴 ∈ V)
10 uniexg 6915 . . . . . . 7 (𝐴𝑉 𝐴 ∈ V)
11 pwsiga 29998 . . . . . . 7 ( 𝐴 ∈ V → 𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴))
1210, 11syl 17 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴))
13 pwuni 4864 . . . . . 6 𝐴 ⊆ 𝒫 𝐴
1412, 13jctir 560 . . . . 5 (𝐴𝑉 → (𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ 𝒫 𝐴))
15 sseq2 3611 . . . . . 6 (𝑠 = 𝒫 𝐴 → (𝐴𝑠𝐴 ⊆ 𝒫 𝐴))
1615elrab 3350 . . . . 5 (𝒫 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ↔ (𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ 𝒫 𝐴))
1714, 16sylibr 224 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
18 ne0i 3902 . . . 4 (𝒫 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅)
1917, 18syl 17 . . 3 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅)
20 intex 4785 . . 3 ({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
2119, 20sylib 208 . 2 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
222, 8, 9, 21fvmptd 6250 1 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  {crab 2911  Vcvv 3189  wss 3559  c0 3896  𝒫 cpw 4135   cuni 4407   cint 4445  cmpt 4678  cfv 5852  sigAlgebracsiga 29975  sigaGencsigagen 30006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-siga 29976  df-sigagen 30007
This theorem is referenced by:  sigagensiga  30009  sssigagen  30013  sigagenss  30017
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