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Theorem salgenval 43404
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 43396 . . 3 SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)})
21a1i 11 . 2 (𝑋𝑉 → SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)}))
3 unieq 4807 . . . . . . 7 (𝑥 = 𝑋 𝑥 = 𝑋)
43eqeq2d 2749 . . . . . 6 (𝑥 = 𝑋 → ( 𝑠 = 𝑥 𝑠 = 𝑋))
5 sseq1 3902 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝑠𝑋𝑠))
64, 5anbi12d 634 . . . . 5 (𝑥 = 𝑋 → (( 𝑠 = 𝑥𝑥𝑠) ↔ ( 𝑠 = 𝑋𝑋𝑠)))
76rabbidv 3381 . . . 4 (𝑥 = 𝑋 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)} = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
87inteqd 4841 . . 3 (𝑥 = 𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)} = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
98adantl 485 . 2 ((𝑋𝑉𝑥 = 𝑋) → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)} = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
10 elex 3416 . 2 (𝑋𝑉𝑋 ∈ V)
11 uniexg 7484 . . . . . . 7 (𝑋𝑉 𝑋 ∈ V)
12 pwsal 43398 . . . . . . 7 ( 𝑋 ∈ V → 𝒫 𝑋 ∈ SAlg)
1311, 12syl 17 . . . . . 6 (𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
14 unipw 5309 . . . . . . 7 𝒫 𝑋 = 𝑋
1514a1i 11 . . . . . 6 (𝑋𝑉 𝒫 𝑋 = 𝑋)
16 pwuni 4835 . . . . . . 7 𝑋 ⊆ 𝒫 𝑋
1716a1i 11 . . . . . 6 (𝑋𝑉𝑋 ⊆ 𝒫 𝑋)
1813, 15, 17jca32 519 . . . . 5 (𝑋𝑉 → (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
19 unieq 4807 . . . . . . . 8 (𝑠 = 𝒫 𝑋 𝑠 = 𝒫 𝑋)
2019eqeq1d 2740 . . . . . . 7 (𝑠 = 𝒫 𝑋 → ( 𝑠 = 𝑋 𝒫 𝑋 = 𝑋))
21 sseq2 3903 . . . . . . 7 (𝑠 = 𝒫 𝑋 → (𝑋𝑠𝑋 ⊆ 𝒫 𝑋))
2220, 21anbi12d 634 . . . . . 6 (𝑠 = 𝒫 𝑋 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
2322elrab 3588 . . . . 5 (𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝒫 𝑋 ∈ SAlg ∧ ( 𝒫 𝑋 = 𝑋𝑋 ⊆ 𝒫 𝑋)))
2418, 23sylibr 237 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
2524ne0d 4224 . . 3 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
26 intex 5205 . . 3 ({𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅ ↔ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ V)
2725, 26sylib 221 . 2 (𝑋𝑉 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ V)
282, 9, 10, 27fvmptd 6782 1 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  wne 2934  {crab 3057  Vcvv 3398  wss 3843  c0 4211  𝒫 cpw 4488   cuni 4796   cint 4836  cmpt 5110  cfv 6339  SAlgcsalg 43391  SalGencsalgen 43395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-int 4837  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-salg 43392  df-salgen 43396
This theorem is referenced by:  salgencl  43413  sssalgen  43416  salgenss  43417  salgenuni  43418  issalgend  43419
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