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Theorem salgenval 43862
Description: The sigma-algebra generated by a set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Assertion
Ref Expression
salgenval (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
Distinct variable group:   𝑋,𝑠
Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem salgenval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-salgen 43854 . . 3 SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)})
21a1i 11 . 2 (𝑋𝑉 → SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)}))
3 unieq 4850 . . . . . . 7 (𝑥 = 𝑋 𝑥 = 𝑋)
43eqeq2d 2749 . . . . . 6 (𝑥 = 𝑋 → ( 𝑠 = 𝑥 𝑠 = 𝑋))
5 sseq1 3946 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝑠𝑋𝑠))
64, 5anbi12d 631 . . . . 5 (𝑥 = 𝑋 → (( 𝑠 = 𝑥𝑥𝑠) ↔ ( 𝑠 = 𝑋𝑋𝑠)))
76rabbidv 3414 . . . 4 (𝑥 = 𝑋 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)} = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
87inteqd 4884 . . 3 (𝑥 = 𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)} = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
98adantl 482 . 2 ((𝑋𝑉𝑥 = 𝑋) → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)} = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
10 elex 3450 . 2 (𝑋𝑉𝑋 ∈ V)
11 uniexg 7593 . . . . . . 7 (𝑋𝑉 𝑋 ∈ V)
12 pwsal 43856 . . . . . . 7 ( 𝑋 ∈ V → 𝒫 𝑋 ∈ SAlg)
1311, 12syl 17 . . . . . 6 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
14 unipw 5366 . . . . . . 7 𝒫 𝑋 = 𝑋
1514a1i 11 . . . . . 6 (𝑋𝑉 𝒫 𝑋 = 𝑋)
16 pwuni 4878 . . . . . . 7 𝑋 ⊆ 𝒫 𝑋
1716a1i 11 . . . . . 6 (𝑋𝑉𝑋 ⊆ 𝒫 𝑋)
1813, 15, 17jca32 516 . . . . 5 (𝑋𝑉 → (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
19 unieq 4850 . . . . . . . 8 (𝑠 = 𝒫 𝑋 𝑠 = 𝒫 𝑋)
2019eqeq1d 2740 . . . . . . 7 (𝑠 = 𝒫 𝑋 → ( 𝑠 = 𝑋 𝒫 𝑋 = 𝑋))
21 sseq2 3947 . . . . . . 7 (𝑠 = 𝒫 𝑋 → (𝑋𝑠𝑋 ⊆ 𝒫 𝑋))
2220, 21anbi12d 631 . . . . . 6 (𝑠 = 𝒫 𝑋 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
2322elrab 3624 . . . . 5 (𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
2418, 23sylibr 233 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
2524ne0d 4269 . . 3 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
26 intex 5261 . . 3 ({𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅ ↔ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ V)
2725, 26sylib 217 . 2 (𝑋𝑉 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ V)
282, 9, 10, 27fvmptd 6882 1 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wne 2943  {crab 3068  Vcvv 3432  wss 3887  c0 4256  𝒫 cpw 4533   cuni 4839   cint 4879  cmpt 5157  cfv 6433  SAlgcsalg 43849  SalGencsalgen 43853
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-salg 43850  df-salgen 43854
This theorem is referenced by:  salgencl  43871  sssalgen  43874  salgenss  43875  salgenuni  43876  issalgend  43877
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