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Theorem salgenss 44730
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 44738, 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 (πœ‘ β†’ 𝑋 ∈ 𝑉)
salgenss.g 𝐺 = (SalGenβ€˜π‘‹)
salgenss.s (πœ‘ β†’ 𝑆 ∈ SAlg)
salgenss.i (πœ‘ β†’ 𝑋 βŠ† 𝑆)
salgenss.u (πœ‘ β†’ βˆͺ 𝑆 = βˆͺ 𝑋)
Assertion
Ref Expression
salgenss (πœ‘ β†’ 𝐺 βŠ† 𝑆)

Proof of Theorem salgenss
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 salgenss.g . . . 4 𝐺 = (SalGenβ€˜π‘‹)
21a1i 11 . . 3 (πœ‘ β†’ 𝐺 = (SalGenβ€˜π‘‹))
3 salgenss.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝑉)
4 salgenval 44715 . . . 4 (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
53, 4syl 17 . . 3 (πœ‘ β†’ (SalGenβ€˜π‘‹) = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
62, 5eqtrd 2771 . 2 (πœ‘ β†’ 𝐺 = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
7 salgenss.s . . . . 5 (πœ‘ β†’ 𝑆 ∈ SAlg)
8 salgenss.u . . . . . 6 (πœ‘ β†’ βˆͺ 𝑆 = βˆͺ 𝑋)
9 salgenss.i . . . . . 6 (πœ‘ β†’ 𝑋 βŠ† 𝑆)
108, 9jca 512 . . . . 5 (πœ‘ β†’ (βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆))
117, 10jca 512 . . . 4 (πœ‘ β†’ (𝑆 ∈ SAlg ∧ (βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆)))
12 unieq 4896 . . . . . . 7 (𝑠 = 𝑆 β†’ βˆͺ 𝑠 = βˆͺ 𝑆)
1312eqeq1d 2733 . . . . . 6 (𝑠 = 𝑆 β†’ (βˆͺ 𝑠 = βˆͺ 𝑋 ↔ βˆͺ 𝑆 = βˆͺ 𝑋))
14 sseq2 3988 . . . . . 6 (𝑠 = 𝑆 β†’ (𝑋 βŠ† 𝑠 ↔ 𝑋 βŠ† 𝑆))
1513, 14anbi12d 631 . . . . 5 (𝑠 = 𝑆 β†’ ((βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠) ↔ (βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆)))
1615elrab 3663 . . . 4 (𝑆 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} ↔ (𝑆 ∈ SAlg ∧ (βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆)))
1711, 16sylibr 233 . . 3 (πœ‘ β†’ 𝑆 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
18 intss1 4944 . . 3 (𝑆 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} βŠ† 𝑆)
1917, 18syl 17 . 2 (πœ‘ β†’ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} βŠ† 𝑆)
206, 19eqsstrd 4000 1 (πœ‘ β†’ 𝐺 βŠ† 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3418   βŠ† wss 3928  βˆͺ cuni 4885  βˆ© cint 4927  β€˜cfv 6516  SAlgcsalg 44702  SalGencsalgen 44706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-iota 6468  df-fun 6518  df-fv 6524  df-salg 44703  df-salgen 44707
This theorem is referenced by:  issalgend  44732  dfsalgen2  44735  borelmbl  45030  smfpimbor1lem2  45193
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