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Theorem salgenss 39230
Description: The sigma-algebra generated by a set is the smallest sigma-algebra, on the same base set, that includes the set. Proposition 111G (b) of [Fremlin1] p. 13. Notice that the condition "on the same base set" is needed, see the counterexample salgensscntex 39238, where a sigma-algebra is shown that includes a set, but does not include the sigma-algebra generated (the key is that its base set is larger than the base set of the generating set). (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
salgenss.x (𝜑𝑋𝑉)
salgenss.g 𝐺 = (SalGen‘𝑋)
salgenss.s (𝜑𝑆 ∈ SAlg)
salgenss.i (𝜑𝑋𝑆)
salgenss.u (𝜑 𝑆 = 𝑋)
Assertion
Ref Expression
salgenss (𝜑𝐺𝑆)

Proof of Theorem salgenss
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 salgenss.g . . . 4 𝐺 = (SalGen‘𝑋)
21a1i 11 . . 3 (𝜑𝐺 = (SalGen‘𝑋))
3 salgenss.x . . . 4 (𝜑𝑋𝑉)
4 salgenval 39217 . . . 4 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
53, 4syl 17 . . 3 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
62, 5eqtrd 2644 . 2 (𝜑𝐺 = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
7 salgenss.s . . . . 5 (𝜑𝑆 ∈ SAlg)
8 salgenss.u . . . . . 6 (𝜑 𝑆 = 𝑋)
9 salgenss.i . . . . . 6 (𝜑𝑋𝑆)
108, 9jca 553 . . . . 5 (𝜑 → ( 𝑆 = 𝑋𝑋𝑆))
117, 10jca 553 . . . 4 (𝜑 → (𝑆 ∈ SAlg ∧ ( 𝑆 = 𝑋𝑋𝑆)))
12 unieq 4380 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
1312eqeq1d 2612 . . . . . 6 (𝑠 = 𝑆 → ( 𝑠 = 𝑋 𝑆 = 𝑋))
14 sseq2 3590 . . . . . 6 (𝑠 = 𝑆 → (𝑋𝑠𝑋𝑆))
1513, 14anbi12d 743 . . . . 5 (𝑠 = 𝑆 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑆 = 𝑋𝑋𝑆)))
1615elrab 3331 . . . 4 (𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑆 ∈ SAlg ∧ ( 𝑆 = 𝑋𝑋𝑆)))
1711, 16sylibr 223 . . 3 (𝜑𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
18 intss1 4427 . . 3 (𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ 𝑆)
1917, 18syl 17 . 2 (𝜑 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ 𝑆)
206, 19eqsstrd 3602 1 (𝜑𝐺𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {crab 2900  wss 3540   cuni 4372   cint 4410  cfv 5804  SAlgcsalg 39204  SalGencsalgen 39208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-salg 39205  df-salgen 39209
This theorem is referenced by:  issalgend  39232  dfsalgen2  39235  borelmbl  39526  smfpimbor1lem2  39684
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