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Theorem salgenss 43875
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 43883, 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 43862 . . . 4 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
53, 4syl 17 . . 3 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
62, 5eqtrd 2778 . 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 512 . . . . 5 (𝜑 → ( 𝑆 = 𝑋𝑋𝑆))
117, 10jca 512 . . . 4 (𝜑 → (𝑆 ∈ SAlg ∧ ( 𝑆 = 𝑋𝑋𝑆)))
12 unieq 4850 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
1312eqeq1d 2740 . . . . . 6 (𝑠 = 𝑆 → ( 𝑠 = 𝑋 𝑆 = 𝑋))
14 sseq2 3947 . . . . . 6 (𝑠 = 𝑆 → (𝑋𝑠𝑋𝑆))
1513, 14anbi12d 631 . . . . 5 (𝑠 = 𝑆 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑆 = 𝑋𝑋𝑆)))
1615elrab 3624 . . . 4 (𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑆 ∈ SAlg ∧ ( 𝑆 = 𝑋𝑋𝑆)))
1711, 16sylibr 233 . . 3 (𝜑𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
18 intss1 4894 . . 3 (𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ 𝑆)
1917, 18syl 17 . 2 (𝜑 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ 𝑆)
206, 19eqsstrd 3959 1 (𝜑𝐺𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  {crab 3068  wss 3887   cuni 4839   cint 4879  cfv 6433  SAlgcsalg 43849  SalGencsalgen 43853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-salg 43850  df-salgen 43854
This theorem is referenced by:  issalgend  43877  dfsalgen2  43880  borelmbl  44174  smfpimbor1lem2  44333
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