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Theorem dmsigagen 32807
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 7680 . . . . . 6 βˆͺ 𝑗 ∈ V
2 pwsiga 32793 . . . . . 6 (βˆͺ 𝑗 ∈ V β†’ 𝒫 βˆͺ 𝑗 ∈ (sigAlgebraβ€˜βˆͺ 𝑗))
31, 2ax-mp 5 . . . . 5 𝒫 βˆͺ 𝑗 ∈ (sigAlgebraβ€˜βˆͺ 𝑗)
4 pwuni 4910 . . . . 5 𝑗 βŠ† 𝒫 βˆͺ 𝑗
5 sseq2 3974 . . . . . 6 (𝑠 = 𝒫 βˆͺ 𝑗 β†’ (𝑗 βŠ† 𝑠 ↔ 𝑗 βŠ† 𝒫 βˆͺ 𝑗))
65rspcev 3583 . . . . 5 ((𝒫 βˆͺ 𝑗 ∈ (sigAlgebraβ€˜βˆͺ 𝑗) ∧ 𝑗 βŠ† 𝒫 βˆͺ 𝑗) β†’ βˆƒπ‘  ∈ (sigAlgebraβ€˜βˆͺ 𝑗)𝑗 βŠ† 𝑠)
73, 4, 6mp2an 691 . . . 4 βˆƒπ‘  ∈ (sigAlgebraβ€˜βˆͺ 𝑗)𝑗 βŠ† 𝑠
8 rabn0 4349 . . . 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 5298 . . 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 32802 . 2 sigaGen = (𝑗 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝑗) ∣ 𝑗 βŠ† 𝑠})
1311, 12dmmpti 6649 1 dom sigaGen = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆƒwrex 3070  {crab 3406  Vcvv 3447   βŠ† wss 3914  βˆ…c0 4286  π’« cpw 4564  βˆͺ cuni 4869  βˆ© cint 4911  dom cdm 5637  β€˜cfv 6500  sigAlgebracsiga 32771  sigaGencsigagen 32801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-siga 32772  df-sigagen 32802
This theorem is referenced by: (None)
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