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Theorem dmsigagen 31302
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 7455 . . . . . 6 𝑗 ∈ V
2 pwsiga 31288 . . . . . 6 ( 𝑗 ∈ V → 𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗))
31, 2ax-mp 5 . . . . 5 𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗)
4 pwuni 4866 . . . . 5 𝑗 ⊆ 𝒫 𝑗
5 sseq2 3990 . . . . . 6 (𝑠 = 𝒫 𝑗 → (𝑗𝑠𝑗 ⊆ 𝒫 𝑗))
65rspcev 3620 . . . . 5 ((𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗) ∧ 𝑗 ⊆ 𝒫 𝑗) → ∃𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠)
73, 4, 6mp2an 688 . . . 4 𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠
8 rabn0 4336 . . . 4 ({𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅ ↔ ∃𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠)
97, 8mpbir 232 . . 3 {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅
10 intex 5231 . . 3 ({𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ∈ V)
119, 10mpbi 231 . 2 {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ∈ V
12 df-sigagen 31297 . 2 sigaGen = (𝑗 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠})
1311, 12dmmpti 6485 1 dom sigaGen = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  wne 3013  wrex 3136  {crab 3139  Vcvv 3492  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830   cint 4867  dom cdm 5548  cfv 6348  sigAlgebracsiga 31266  sigaGencsigagen 31296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-siga 31267  df-sigagen 31297
This theorem is referenced by: (None)
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