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Theorem sigagenval 30543
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 30542 . . 3 sigaGen = (𝑥 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠})
21a1i 11 . 2 (𝐴𝑉 → sigaGen = (𝑥 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠}))
3 unieq 4582 . . . . . 6 (𝑥 = 𝐴 𝑥 = 𝐴)
43fveq2d 6336 . . . . 5 (𝑥 = 𝐴 → (sigAlgebra‘ 𝑥) = (sigAlgebra‘ 𝐴))
5 sseq1 3775 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑠𝐴𝑠))
64, 5rabeqbidv 3345 . . . 4 (𝑥 = 𝐴 → {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
76inteqd 4616 . . 3 (𝑥 = 𝐴 {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
87adantl 467 . 2 ((𝐴𝑉𝑥 = 𝐴) → {𝑠 ∈ (sigAlgebra‘ 𝑥) ∣ 𝑥𝑠} = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
9 elex 3363 . 2 (𝐴𝑉𝐴 ∈ V)
10 uniexg 7102 . . . . . . 7 (𝐴𝑉 𝐴 ∈ V)
11 pwsiga 30533 . . . . . . 7 ( 𝐴 ∈ V → 𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴))
1210, 11syl 17 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴))
13 pwuni 4610 . . . . . 6 𝐴 ⊆ 𝒫 𝐴
1412, 13jctir 504 . . . . 5 (𝐴𝑉 → (𝒫 𝐴 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ 𝒫 𝐴))
15 sseq2 3776 . . . . . 6 (𝑠 = 𝒫 𝐴 → (𝐴𝑠𝐴 ⊆ 𝒫 𝐴))
1615elrab 3515 . . . . 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 4069 . . . 4 (𝒫 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅)
1917, 18syl 17 . . 3 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅)
20 intex 4951 . . 3 ({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
2119, 20sylib 208 . 2 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
222, 8, 9, 21fvmptd 6430 1 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  {crab 3065  Vcvv 3351  wss 3723  c0 4063  𝒫 cpw 4297   cuni 4574   cint 4611  cmpt 4863  cfv 6031  sigAlgebracsiga 30510  sigaGencsigagen 30541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-siga 30511  df-sigagen 30542
This theorem is referenced by:  sigagensiga  30544  sssigagen  30548  sigagenss  30552
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