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Theorem sigagensiga 29977
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 29976 . 2 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
2 fvex 6160 . . . . 5 (sigaGen‘𝐴) ∈ V
31, 2syl6eqelr 2713 . . . 4 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
4 intex 4785 . . . 4 ({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
53, 4sylibr 224 . . 3 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅)
6 ssrab2 3671 . . . . 5 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ (sigAlgebra‘ 𝐴)
76a1i 11 . . . 4 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ (sigAlgebra‘ 𝐴))
8 fvex 6160 . . . . 5 (sigAlgebra‘ 𝐴) ∈ V
98elpw2 4793 . . . 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 29973 . . 3 (({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ∧ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ 𝒫 (sigAlgebra‘ 𝐴)) → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ (sigAlgebra‘ 𝐴))
125, 10, 11syl2anc 692 . 2 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ (sigAlgebra‘ 𝐴))
131, 12eqeltrd 2704 1 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1992  wne 2796  {crab 2916  Vcvv 3191  wss 3560  c0 3896  𝒫 cpw 4135   cuni 4407   cint 4445  cfv 5850  sigAlgebracsiga 29943  sigaGencsigagen 29974
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  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 5813  df-fun 5852  df-fv 5858  df-siga 29944  df-sigagen 29975
This theorem is referenced by:  sgsiga  29978  unisg  29979  sigagenss2  29986  brsiga  30019  brsigarn  30020  cldssbrsiga  30023  sxsiga  30027  cnmbfm  30098  sxbrsiga  30125
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