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Theorem salgenval 39845
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 39837 . . 3 SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)})
21a1i 11 . 2 (𝑋𝑉 → SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)}))
3 unieq 4410 . . . . . . 7 (𝑥 = 𝑋 𝑥 = 𝑋)
43eqeq2d 2631 . . . . . 6 (𝑥 = 𝑋 → ( 𝑠 = 𝑥 𝑠 = 𝑋))
5 sseq1 3605 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝑠𝑋𝑠))
64, 5anbi12d 746 . . . . 5 (𝑥 = 𝑋 → (( 𝑠 = 𝑥𝑥𝑠) ↔ ( 𝑠 = 𝑋𝑋𝑠)))
76rabbidv 3177 . . . 4 (𝑥 = 𝑋 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)} = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
87inteqd 4445 . . 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 3198 . 2 (𝑋𝑉𝑋 ∈ V)
11 uniexg 6908 . . . . . . 7 (𝑋𝑉 𝑋 ∈ V)
12 pwsal 39839 . . . . . . 7 ( 𝑋 ∈ V → 𝒫 𝑋 ∈ SAlg)
1311, 12syl 17 . . . . . 6 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
14 unipw 4879 . . . . . . 7 𝒫 𝑋 = 𝑋
1514a1i 11 . . . . . 6 (𝑋𝑉 𝒫 𝑋 = 𝑋)
16 pwuni 4859 . . . . . . 7 𝑋 ⊆ 𝒫 𝑋
1716a1i 11 . . . . . 6 (𝑋𝑉𝑋 ⊆ 𝒫 𝑋)
1813, 15, 17jca32 557 . . . . 5 (𝑋𝑉 → (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
19 unieq 4410 . . . . . . . 8 (𝑠 = 𝒫 𝑋 𝑠 = 𝒫 𝑋)
2019eqeq1d 2623 . . . . . . 7 (𝑠 = 𝒫 𝑋 → ( 𝑠 = 𝑋 𝒫 𝑋 = 𝑋))
21 sseq2 3606 . . . . . . 7 (𝑠 = 𝒫 𝑋 → (𝑋𝑠𝑋 ⊆ 𝒫 𝑋))
2220, 21anbi12d 746 . . . . . 6 (𝑠 = 𝒫 𝑋 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
2322elrab 3346 . . . . 5 (𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
2418, 23sylibr 224 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
25 ne0i 3897 . . . 4 (𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
2624, 25syl 17 . . 3 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
27 intex 4780 . . 3 ({𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅ ↔ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ V)
2826, 27sylib 208 . 2 (𝑋𝑉 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ V)
292, 9, 10, 28fvmptd 6245 1 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  {crab 2911  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402   cint 4440  cmpt 4673  cfv 5847  SAlgcsalg 39832  SalGencsalgen 39836
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  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 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-salg 39833  df-salgen 39837
This theorem is referenced by:  salgencl  39854  sssalgen  39857  salgenss  39858  salgenuni  39859  issalgend  39860
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