Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sigagensiga Structured version   Visualization version   GIF version

Theorem sigagensiga 34143
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 34142 . 2 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
2 fvex 6918 . . . . 5 (sigaGen‘𝐴) ∈ V
31, 2eqeltrrdi 2849 . . . 4 (𝐴𝑉 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
4 intex 5343 . . . 4 ({𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ∈ V)
53, 4sylibr 234 . . 3 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ≠ ∅)
6 ssrab2 4079 . . . . 5 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ (sigAlgebra‘ 𝐴)
76a1i 11 . . . 4 (𝐴𝑉 → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ (sigAlgebra‘ 𝐴))
8 fvex 6918 . . . . 5 (sigAlgebra‘ 𝐴) ∈ V
98elpw2 5333 . . . 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 34139 . . 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 2840 1 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wne 2939  {crab 3435  Vcvv 3479  wss 3950  c0 4332  𝒫 cpw 4599   cuni 4906   cint 4945  cfv 6560  sigAlgebracsiga 34110  sigaGencsigagen 34140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-siga 34111  df-sigagen 34141
This theorem is referenced by:  sgsiga  34144  unisg  34145  sigagenss2  34152  brsiga  34185  brsigarn  34186  cldssbrsiga  34189  sxsiga  34193  cnmbfm  34266  sxbrsiga  34293
  Copyright terms: Public domain W3C validator