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Theorem salgencld 40330
Description: SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
salgencld.1 (𝜑𝑋𝑉)
salgencld.2 𝑆 = (SalGen‘𝑋)
Assertion
Ref Expression
salgencld (𝜑𝑆 ∈ SAlg)

Proof of Theorem salgencld
StepHypRef Expression
1 salgencld.2 . 2 𝑆 = (SalGen‘𝑋)
2 salgencld.1 . . 3 (𝜑𝑋𝑉)
3 salgencl 40313 . . 3 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
42, 3syl 17 . 2 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
51, 4syl5eqel 2703 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  cfv 5876  SAlgcsalg 40291  SalGencsalgen 40295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-salg 40292  df-salgen 40296
This theorem is referenced by:  bor1sal  40336  cnfsmf  40712  incsmf  40714  bormflebmf  40725  decsmf  40738  smf2id  40771  smfco  40772
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