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Theorem sigagensiga 30332
Description: A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
sigagensiga (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))

Proof of Theorem sigagensiga
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 sigagenval 30331 . 2 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
2 fvex 6239 . . . . 5 (sigaGen‘𝐴) ∈ V
31, 2syl6eqelr 2739 . . . 4 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
4 intex 4850 . . . 4 ({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
53, 4sylibr 224 . . 3 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅)
6 ssrab2 3720 . . . . 5 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ (sigAlgebra‘ 𝐴)
76a1i 11 . . . 4 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ (sigAlgebra‘ 𝐴))
8 fvex 6239 . . . . 5 (sigAlgebra‘ 𝐴) ∈ V
98elpw2 4858 . . . 4 ({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ 𝒫 (sigAlgebra‘ 𝐴) ↔ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ (sigAlgebra‘ 𝐴))
107, 9sylibr 224 . . 3 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ 𝒫 (sigAlgebra‘ 𝐴))
11 insiga 30328 . . 3 (({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ∧ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ 𝒫 (sigAlgebra‘ 𝐴)) → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ (sigAlgebra‘ 𝐴))
125, 10, 11syl2anc 694 . 2 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ (sigAlgebra‘ 𝐴))
131, 12eqeltrd 2730 1 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wne 2823  {crab 2945  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468   cint 4507  cfv 5926  sigAlgebracsiga 30298  sigaGencsigagen 30329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-siga 30299  df-sigagen 30330
This theorem is referenced by:  sgsiga  30333  unisg  30334  sigagenss2  30341  brsiga  30374  brsigarn  30375  cldssbrsiga  30378  sxsiga  30382  cnmbfm  30453  sxbrsiga  30480
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