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Theorem dmsigagen 34125
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 7758 . . . . . 6 𝑗 ∈ V
2 pwsiga 34111 . . . . . 6 ( 𝑗 ∈ V → 𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗))
31, 2ax-mp 5 . . . . 5 𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗)
4 pwuni 4950 . . . . 5 𝑗 ⊆ 𝒫 𝑗
5 sseq2 4022 . . . . . 6 (𝑠 = 𝒫 𝑗 → (𝑗𝑠𝑗 ⊆ 𝒫 𝑗))
65rspcev 3622 . . . . 5 ((𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗) ∧ 𝑗 ⊆ 𝒫 𝑗) → ∃𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠)
73, 4, 6mp2an 692 . . . 4 𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠
8 rabn0 4395 . . . 4 ({𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅ ↔ ∃𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠)
97, 8mpbir 231 . . 3 {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅
10 intex 5350 . . 3 ({𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ∈ V)
119, 10mpbi 230 . 2 {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ∈ V
12 df-sigagen 34120 . 2 sigaGen = (𝑗 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠})
1311, 12dmmpti 6713 1 dom sigaGen = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912   cint 4951  dom cdm 5689  cfv 6563  sigAlgebracsiga 34089  sigaGencsigagen 34119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-siga 34090  df-sigagen 34120
This theorem is referenced by: (None)
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