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Theorem salgenss 41345
 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 41353, 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 41332 . . . 4 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
53, 4syl 17 . . 3 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
62, 5eqtrd 2861 . 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 509 . . . . 5 (𝜑 → ( 𝑆 = 𝑋𝑋𝑆))
117, 10jca 509 . . . 4 (𝜑 → (𝑆 ∈ SAlg ∧ ( 𝑆 = 𝑋𝑋𝑆)))
12 unieq 4666 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
1312eqeq1d 2827 . . . . . 6 (𝑠 = 𝑆 → ( 𝑠 = 𝑋 𝑆 = 𝑋))
14 sseq2 3852 . . . . . 6 (𝑠 = 𝑆 → (𝑋𝑠𝑋𝑆))
1513, 14anbi12d 626 . . . . 5 (𝑠 = 𝑆 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑆 = 𝑋𝑋𝑆)))
1615elrab 3585 . . . 4 (𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑆 ∈ SAlg ∧ ( 𝑆 = 𝑋𝑋𝑆)))
1711, 16sylibr 226 . . 3 (𝜑𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
18 intss1 4712 . . 3 (𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ 𝑆)
1917, 18syl 17 . 2 (𝜑 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ 𝑆)
206, 19eqsstrd 3864 1 (𝜑𝐺𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  {crab 3121   ⊆ wss 3798  ∪ cuni 4658  ∩ cint 4697  ‘cfv 6123  SAlgcsalg 41319  SalGencsalgen 41323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-salg 41320  df-salgen 41324 This theorem is referenced by:  issalgend  41347  dfsalgen2  41350  borelmbl  41644  smfpimbor1lem2  41800
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