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Theorem salgenss 42487
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 42495, 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 42474 . . . 4 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
53, 4syl 17 . . 3 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
62, 5eqtrd 2860 . 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 4844 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
1312eqeq1d 2827 . . . . . 6 (𝑠 = 𝑆 → ( 𝑠 = 𝑋 𝑆 = 𝑋))
14 sseq2 3996 . . . . . 6 (𝑠 = 𝑆 → (𝑋𝑠𝑋𝑆))
1513, 14anbi12d 630 . . . . 5 (𝑠 = 𝑆 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑆 = 𝑋𝑋𝑆)))
1615elrab 3683 . . . 4 (𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑆 ∈ SAlg ∧ ( 𝑆 = 𝑋𝑋𝑆)))
1711, 16sylibr 235 . . 3 (𝜑𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
18 intss1 4888 . . 3 (𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ 𝑆)
1917, 18syl 17 . 2 (𝜑 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ 𝑆)
206, 19eqsstrd 4008 1 (𝜑𝐺𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  {crab 3146  wss 3939   cuni 4836   cint 4873  cfv 6351  SAlgcsalg 42461  SalGencsalgen 42465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-int 4874  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fv 6359  df-salg 42462  df-salgen 42466
This theorem is referenced by:  issalgend  42489  dfsalgen2  42492  borelmbl  42786  smfpimbor1lem2  42942
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