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Theorem dmsigagen 30547
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 7105 . . . . . 6 𝑗 ∈ V
2 pwsiga 30533 . . . . . 6 ( 𝑗 ∈ V → 𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗))
31, 2ax-mp 5 . . . . 5 𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗)
4 pwuni 4611 . . . . 5 𝑗 ⊆ 𝒫 𝑗
5 sseq2 3776 . . . . . 6 (𝑠 = 𝒫 𝑗 → (𝑗𝑠𝑗 ⊆ 𝒫 𝑗))
65rspcev 3460 . . . . 5 ((𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗) ∧ 𝑗 ⊆ 𝒫 𝑗) → ∃𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠)
73, 4, 6mp2an 672 . . . 4 𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠
8 rabn0 4105 . . . 4 ({𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅ ↔ ∃𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠)
97, 8mpbir 221 . . 3 {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅
10 intex 4952 . . 3 ({𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ∈ V)
119, 10mpbi 220 . 2 {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ∈ V
12 df-sigagen 30542 . 2 sigaGen = (𝑗 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠})
1311, 12dmmpti 6162 1 dom sigaGen = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  wne 2943  wrex 3062  {crab 3065  Vcvv 3351  wss 3723  c0 4063  𝒫 cpw 4298   cuni 4575   cint 4612  dom cdm 5250  cfv 6030  sigAlgebracsiga 30510  sigaGencsigagen 30541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fn 6033  df-fv 6038  df-siga 30511  df-sigagen 30542
This theorem is referenced by: (None)
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