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Theorem dmsigagen 30000
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 6910 . . . . . 6 𝑗 ∈ V
2 pwsiga 29986 . . . . . 6 ( 𝑗 ∈ V → 𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗))
31, 2ax-mp 5 . . . . 5 𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗)
4 pwuni 4861 . . . . 5 𝑗 ⊆ 𝒫 𝑗
5 sseq2 3608 . . . . . 6 (𝑠 = 𝒫 𝑗 → (𝑗𝑠𝑗 ⊆ 𝒫 𝑗))
65rspcev 3295 . . . . 5 ((𝒫 𝑗 ∈ (sigAlgebra‘ 𝑗) ∧ 𝑗 ⊆ 𝒫 𝑗) → ∃𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠)
73, 4, 6mp2an 707 . . . 4 𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠
8 rabn0 3934 . . . 4 ({𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅ ↔ ∃𝑠 ∈ (sigAlgebra‘ 𝑗)𝑗𝑠)
97, 8mpbir 221 . . 3 {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅
10 intex 4782 . . 3 ({𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ≠ ∅ ↔ {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ∈ V)
119, 10mpbi 220 . 2 {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠} ∈ V
12 df-sigagen 29995 . 2 sigaGen = (𝑗 ∈ V ↦ {𝑠 ∈ (sigAlgebra‘ 𝑗) ∣ 𝑗𝑠})
1311, 12dmmpti 5982 1 dom sigaGen = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  wne 2790  wrex 2908  {crab 2911  Vcvv 3186  wss 3556  c0 3893  𝒫 cpw 4132   cuni 4404   cint 4442  dom cdm 5076  cfv 5849  sigAlgebracsiga 29963  sigaGencsigagen 29994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-int 4443  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-iota 5812  df-fun 5851  df-fn 5852  df-fv 5857  df-siga 29964  df-sigagen 29995
This theorem is referenced by: (None)
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