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 33137
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 7728 . . . . . 6 βˆͺ 𝑗 ∈ V
2 pwsiga 33123 . . . . . 6 (βˆͺ 𝑗 ∈ V β†’ 𝒫 βˆͺ 𝑗 ∈ (sigAlgebraβ€˜βˆͺ 𝑗))
31, 2ax-mp 5 . . . . 5 𝒫 βˆͺ 𝑗 ∈ (sigAlgebraβ€˜βˆͺ 𝑗)
4 pwuni 4949 . . . . 5 𝑗 βŠ† 𝒫 βˆͺ 𝑗
5 sseq2 4008 . . . . . 6 (𝑠 = 𝒫 βˆͺ 𝑗 β†’ (𝑗 βŠ† 𝑠 ↔ 𝑗 βŠ† 𝒫 βˆͺ 𝑗))
65rspcev 3612 . . . . 5 ((𝒫 βˆͺ 𝑗 ∈ (sigAlgebraβ€˜βˆͺ 𝑗) ∧ 𝑗 βŠ† 𝒫 βˆͺ 𝑗) β†’ βˆƒπ‘  ∈ (sigAlgebraβ€˜βˆͺ 𝑗)𝑗 βŠ† 𝑠)
73, 4, 6mp2an 690 . . . 4 βˆƒπ‘  ∈ (sigAlgebraβ€˜βˆͺ 𝑗)𝑗 βŠ† 𝑠
8 rabn0 4385 . . . 4 ({𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝑗) ∣ 𝑗 βŠ† 𝑠} β‰  βˆ… ↔ βˆƒπ‘  ∈ (sigAlgebraβ€˜βˆͺ 𝑗)𝑗 βŠ† 𝑠)
97, 8mpbir 230 . . 3 {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝑗) ∣ 𝑗 βŠ† 𝑠} β‰  βˆ…
10 intex 5337 . . 3 ({𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝑗) ∣ 𝑗 βŠ† 𝑠} β‰  βˆ… ↔ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝑗) ∣ 𝑗 βŠ† 𝑠} ∈ V)
119, 10mpbi 229 . 2 ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝑗) ∣ 𝑗 βŠ† 𝑠} ∈ V
12 df-sigagen 33132 . 2 sigaGen = (𝑗 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝑗) ∣ 𝑗 βŠ† 𝑠})
1311, 12dmmpti 6694 1 dom sigaGen = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  βˆͺ cuni 4908  βˆ© cint 4950  dom cdm 5676  β€˜cfv 6543  sigAlgebracsiga 33101  sigaGencsigagen 33131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-siga 33102  df-sigagen 33132
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator