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Theorem salgenval 42475
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 42467 . . 3 SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)})
21a1i 11 . 2 (𝑋𝑉 → SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)}))
3 unieq 4845 . . . . . . 7 (𝑥 = 𝑋 𝑥 = 𝑋)
43eqeq2d 2837 . . . . . 6 (𝑥 = 𝑋 → ( 𝑠 = 𝑥 𝑠 = 𝑋))
5 sseq1 3996 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝑠𝑋𝑠))
64, 5anbi12d 630 . . . . 5 (𝑥 = 𝑋 → (( 𝑠 = 𝑥𝑥𝑠) ↔ ( 𝑠 = 𝑋𝑋𝑠)))
76rabbidv 3486 . . . 4 (𝑥 = 𝑋 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)} = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
87inteqd 4879 . . 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 3518 . 2 (𝑋𝑉𝑋 ∈ V)
11 uniexg 7457 . . . . . . 7 (𝑋𝑉 𝑋 ∈ V)
12 pwsal 42469 . . . . . . 7 ( 𝑋 ∈ V → 𝒫 𝑋 ∈ SAlg)
1311, 12syl 17 . . . . . 6 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
14 unipw 5339 . . . . . . 7 𝒫 𝑋 = 𝑋
1514a1i 11 . . . . . 6 (𝑋𝑉 𝒫 𝑋 = 𝑋)
16 pwuni 4873 . . . . . . 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 4845 . . . . . . . 8 (𝑠 = 𝒫 𝑋 𝑠 = 𝒫 𝑋)
2019eqeq1d 2828 . . . . . . 7 (𝑠 = 𝒫 𝑋 → ( 𝑠 = 𝑋 𝒫 𝑋 = 𝑋))
21 sseq2 3997 . . . . . . 7 (𝑠 = 𝒫 𝑋 → (𝑋𝑠𝑋 ⊆ 𝒫 𝑋))
2220, 21anbi12d 630 . . . . . 6 (𝑠 = 𝒫 𝑋 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
2322elrab 3684 . . . . 5 (𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
2418, 23sylibr 235 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
2524ne0d 4305 . . 3 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
26 intex 5237 . . 3 ({𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅ ↔ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ V)
2725, 26sylib 219 . 2 (𝑋𝑉 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ V)
282, 9, 10, 27fvmptd 6771 1 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wne 3021  {crab 3147  Vcvv 3500  wss 3940  c0 4295  𝒫 cpw 4542   cuni 4837   cint 4874  cmpt 5143  cfv 6352  SAlgcsalg 42462  SalGencsalgen 42466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6312  df-fun 6354  df-fv 6360  df-salg 42463  df-salgen 42467
This theorem is referenced by:  salgencl  42484  sssalgen  42487  salgenss  42488  salgenuni  42489  issalgend  42490
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