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Theorem salgenval 41059
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 41051 . . 3 SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)})
21a1i 11 . 2 (𝑋𝑉 → SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)}))
3 unieq 4583 . . . . . . 7 (𝑥 = 𝑋 𝑥 = 𝑋)
43eqeq2d 2781 . . . . . 6 (𝑥 = 𝑋 → ( 𝑠 = 𝑥 𝑠 = 𝑋))
5 sseq1 3776 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝑠𝑋𝑠))
64, 5anbi12d 610 . . . . 5 (𝑥 = 𝑋 → (( 𝑠 = 𝑥𝑥𝑠) ↔ ( 𝑠 = 𝑋𝑋𝑠)))
76rabbidv 3339 . . . 4 (𝑥 = 𝑋 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)} = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
87inteqd 4617 . . 3 (𝑥 = 𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)} = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
98adantl 467 . 2 ((𝑋𝑉𝑥 = 𝑋) → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)} = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
10 elex 3364 . 2 (𝑋𝑉𝑋 ∈ V)
11 uniexg 7103 . . . . . . 7 (𝑋𝑉 𝑋 ∈ V)
12 pwsal 41053 . . . . . . 7 ( 𝑋 ∈ V → 𝒫 𝑋 ∈ SAlg)
1311, 12syl 17 . . . . . 6 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
14 unipw 5047 . . . . . . 7 𝒫 𝑋 = 𝑋
1514a1i 11 . . . . . 6 (𝑋𝑉 𝒫 𝑋 = 𝑋)
16 pwuni 4611 . . . . . . 7 𝑋 ⊆ 𝒫 𝑋
1716a1i 11 . . . . . 6 (𝑋𝑉𝑋 ⊆ 𝒫 𝑋)
1813, 15, 17jca32 501 . . . . 5 (𝑋𝑉 → (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
19 unieq 4583 . . . . . . . 8 (𝑠 = 𝒫 𝑋 𝑠 = 𝒫 𝑋)
2019eqeq1d 2773 . . . . . . 7 (𝑠 = 𝒫 𝑋 → ( 𝑠 = 𝑋 𝒫 𝑋 = 𝑋))
21 sseq2 3777 . . . . . . 7 (𝑠 = 𝒫 𝑋 → (𝑋𝑠𝑋 ⊆ 𝒫 𝑋))
2220, 21anbi12d 610 . . . . . 6 (𝑠 = 𝒫 𝑋 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
2322elrab 3516 . . . . 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 4070 . . . 4 (𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
2624, 25syl 17 . . 3 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
27 intex 4952 . . 3 ({𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅ ↔ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ V)
2826, 27sylib 208 . 2 (𝑋𝑉 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ V)
292, 9, 10, 28fvmptd 6431 1 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  {crab 3065  Vcvv 3351  wss 3724  c0 4064  𝒫 cpw 4298   cuni 4575   cint 4612  cmpt 4864  cfv 6032  SAlgcsalg 41046  SalGencsalgen 41050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5995  df-fun 6034  df-fv 6040  df-salg 41047  df-salgen 41051
This theorem is referenced by:  salgencl  41068  sssalgen  41071  salgenss  41072  salgenuni  41073  issalgend  41074
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