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Theorem salgencld 42633
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 42616 . . 3 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
42, 3syl 17 . 2 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
51, 4eqeltrid 2917 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  cfv 6354  SAlgcsalg 42594  SalGencsalgen 42598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-int 4876  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-salg 42595  df-salgen 42599
This theorem is referenced by:  bor1sal  42639  cnfsmf  43018  incsmf  43020  bormflebmf  43031  decsmf  43044  smf2id  43077  smfco  43078
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