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Theorem salgenss 43417
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 43425, 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 43404 . . . 4 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
53, 4syl 17 . . 3 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
62, 5eqtrd 2773 . 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 515 . . . . 5 (𝜑 → ( 𝑆 = 𝑋𝑋𝑆))
117, 10jca 515 . . . 4 (𝜑 → (𝑆 ∈ SAlg ∧ ( 𝑆 = 𝑋𝑋𝑆)))
12 unieq 4807 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
1312eqeq1d 2740 . . . . . 6 (𝑠 = 𝑆 → ( 𝑠 = 𝑋 𝑆 = 𝑋))
14 sseq2 3903 . . . . . 6 (𝑠 = 𝑆 → (𝑋𝑠𝑋𝑆))
1513, 14anbi12d 634 . . . . 5 (𝑠 = 𝑆 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑆 = 𝑋𝑋𝑆)))
1615elrab 3588 . . . 4 (𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑆 ∈ SAlg ∧ ( 𝑆 = 𝑋𝑋𝑆)))
1711, 16sylibr 237 . . 3 (𝜑𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
18 intss1 4851 . . 3 (𝑆 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ 𝑆)
1917, 18syl 17 . 2 (𝜑 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ 𝑆)
206, 19eqsstrd 3915 1 (𝜑𝐺𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  {crab 3057  wss 3843   cuni 4796   cint 4836  cfv 6339  SAlgcsalg 43391  SalGencsalgen 43395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-int 4837  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-salg 43392  df-salgen 43396
This theorem is referenced by:  issalgend  43419  dfsalgen2  43422  borelmbl  43716  smfpimbor1lem2  43872
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