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Theorem salgenval 44636
Description: The sigma-algebra generated by a set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Assertion
Ref Expression
salgenval (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
Distinct variable group:   𝑋,𝑠
Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem salgenval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 df-salgen 44628 . . 3 SalGen = (π‘₯ ∈ V ↦ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ π‘₯ ∧ π‘₯ βŠ† 𝑠)})
21a1i 11 . 2 (𝑋 ∈ 𝑉 β†’ SalGen = (π‘₯ ∈ V ↦ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ π‘₯ ∧ π‘₯ βŠ† 𝑠)}))
3 unieq 4881 . . . . . . 7 (π‘₯ = 𝑋 β†’ βˆͺ π‘₯ = βˆͺ 𝑋)
43eqeq2d 2748 . . . . . 6 (π‘₯ = 𝑋 β†’ (βˆͺ 𝑠 = βˆͺ π‘₯ ↔ βˆͺ 𝑠 = βˆͺ 𝑋))
5 sseq1 3974 . . . . . 6 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† 𝑠 ↔ 𝑋 βŠ† 𝑠))
64, 5anbi12d 632 . . . . 5 (π‘₯ = 𝑋 β†’ ((βˆͺ 𝑠 = βˆͺ π‘₯ ∧ π‘₯ βŠ† 𝑠) ↔ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)))
76rabbidv 3418 . . . 4 (π‘₯ = 𝑋 β†’ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ π‘₯ ∧ π‘₯ βŠ† 𝑠)} = {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
87inteqd 4917 . . 3 (π‘₯ = 𝑋 β†’ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ π‘₯ ∧ π‘₯ βŠ† 𝑠)} = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
98adantl 483 . 2 ((𝑋 ∈ 𝑉 ∧ π‘₯ = 𝑋) β†’ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ π‘₯ ∧ π‘₯ βŠ† 𝑠)} = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
10 elex 3466 . 2 (𝑋 ∈ 𝑉 β†’ 𝑋 ∈ V)
11 uniexg 7682 . . . . . . 7 (𝑋 ∈ 𝑉 β†’ βˆͺ 𝑋 ∈ V)
12 pwsal 44630 . . . . . . 7 (βˆͺ 𝑋 ∈ V β†’ 𝒫 βˆͺ 𝑋 ∈ SAlg)
1311, 12syl 17 . . . . . 6 (𝑋 ∈ 𝑉 β†’ 𝒫 βˆͺ 𝑋 ∈ SAlg)
14 unipw 5412 . . . . . . 7 βˆͺ 𝒫 βˆͺ 𝑋 = βˆͺ 𝑋
1514a1i 11 . . . . . 6 (𝑋 ∈ 𝑉 β†’ βˆͺ 𝒫 βˆͺ 𝑋 = βˆͺ 𝑋)
16 pwuni 4911 . . . . . . 7 𝑋 βŠ† 𝒫 βˆͺ 𝑋
1716a1i 11 . . . . . 6 (𝑋 ∈ 𝑉 β†’ 𝑋 βŠ† 𝒫 βˆͺ 𝑋)
1813, 15, 17jca32 517 . . . . 5 (𝑋 ∈ 𝑉 β†’ (𝒫 βˆͺ 𝑋 ∈ SAlg ∧ (βˆͺ 𝒫 βˆͺ 𝑋 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝒫 βˆͺ 𝑋)))
19 unieq 4881 . . . . . . . 8 (𝑠 = 𝒫 βˆͺ 𝑋 β†’ βˆͺ 𝑠 = βˆͺ 𝒫 βˆͺ 𝑋)
2019eqeq1d 2739 . . . . . . 7 (𝑠 = 𝒫 βˆͺ 𝑋 β†’ (βˆͺ 𝑠 = βˆͺ 𝑋 ↔ βˆͺ 𝒫 βˆͺ 𝑋 = βˆͺ 𝑋))
21 sseq2 3975 . . . . . . 7 (𝑠 = 𝒫 βˆͺ 𝑋 β†’ (𝑋 βŠ† 𝑠 ↔ 𝑋 βŠ† 𝒫 βˆͺ 𝑋))
2220, 21anbi12d 632 . . . . . 6 (𝑠 = 𝒫 βˆͺ 𝑋 β†’ ((βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠) ↔ (βˆͺ 𝒫 βˆͺ 𝑋 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝒫 βˆͺ 𝑋)))
2322elrab 3650 . . . . 5 (𝒫 βˆͺ 𝑋 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} ↔ (𝒫 βˆͺ 𝑋 ∈ SAlg ∧ (βˆͺ 𝒫 βˆͺ 𝑋 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝒫 βˆͺ 𝑋)))
2418, 23sylibr 233 . . . 4 (𝑋 ∈ 𝑉 β†’ 𝒫 βˆͺ 𝑋 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
2524ne0d 4300 . . 3 (𝑋 ∈ 𝑉 β†’ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β‰  βˆ…)
26 intex 5299 . . 3 ({𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β‰  βˆ… ↔ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} ∈ V)
2725, 26sylib 217 . 2 (𝑋 ∈ 𝑉 β†’ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} ∈ V)
282, 9, 10, 27fvmptd 6960 1 (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  {crab 3410  Vcvv 3448   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565  βˆͺ cuni 4870  βˆ© cint 4912   ↦ cmpt 5193  β€˜cfv 6501  SAlgcsalg 44623  SalGencsalgen 44627
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-salg 44624  df-salgen 44628
This theorem is referenced by:  salgencl  44647  sssalgen  44650  salgenss  44651  salgenuni  44652  issalgend  44653
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