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

Theorem dmsigagen 34146
Description: A sigma-algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Assertion
Ref Expression
dmsigagen dom sigaGen = V

Proof of Theorem dmsigagen
Dummy variables 𝑗 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vuniex 7760 . . . . . 6 𝑗 ∈ V
2 pwsiga 34132 . . . . . 6 ( 𝑗 ∈ V → 𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗))
31, 2ax-mp 5 . . . . 5 𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗)
4 pwuni 4944 . . . . 5 𝑗 ⊆ 𝒫 𝑗
5 sseq2 4009 . . . . . 6 (𝑠 = 𝒫 𝑗 → (𝑗𝑠𝑗 ⊆ 𝒫 𝑗))
65rspcev 3621 . . . . 5 ((𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗) ∧ 𝑗 ⊆ 𝒫 𝑗) → ∃𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠)
73, 4, 6mp2an 692 . . . 4 𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠
8 rabn0 4388 . . . 4 ({𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅ ↔ ∃𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠)
97, 8mpbir 231 . . 3 {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅
10 intex 5343 . . 3 ({𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ∈ V)
119, 10mpbi 230 . 2 {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ∈ V
12 df-sigagen 34141 . 2 sigaGen = (𝑗 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠})
1311, 12dmmpti 6711 1 dom sigaGen = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  wne 2939  wrex 3069  {crab 3435  Vcvv 3479  wss 3950  c0 4332  𝒫 cpw 4599   cuni 4906   cint 4945  dom cdm 5684  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-fn 6563  df-fv 6568  df-siga 34111  df-sigagen 34141
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator