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Theorem sigagensiga 34122
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 34121 . 2 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
2 fvex 6920 . . . . 5 (sigaGen‘𝐴) ∈ V
31, 2eqeltrrdi 2848 . . . 4 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
4 intex 5350 . . . 4 ({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
53, 4sylibr 234 . . 3 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅)
6 ssrab2 4090 . . . . 5 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ (sigAlgebra‘ 𝐴)
76a1i 11 . . . 4 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ (sigAlgebra‘ 𝐴))
8 fvex 6920 . . . . 5 (sigAlgebra‘ 𝐴) ∈ V
98elpw2 5340 . . . 4 ({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ 𝒫 (sigAlgebra‘ 𝐴) ↔ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ (sigAlgebra‘ 𝐴))
107, 9sylibr 234 . . 3 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ 𝒫 (sigAlgebra‘ 𝐴))
11 insiga 34118 . . 3 (({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ∧ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ 𝒫 (sigAlgebra‘ 𝐴)) → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ (sigAlgebra‘ 𝐴))
125, 10, 11syl2anc 584 . 2 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ (sigAlgebra‘ 𝐴))
131, 12eqeltrd 2839 1 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2938  {crab 3433  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912   cint 4951  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:  sgsiga  34123  unisg  34124  sigagenss2  34131  brsiga  34164  brsigarn  34165  cldssbrsiga  34168  sxsiga  34172  cnmbfm  34245  sxbrsiga  34272
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