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Theorem salgencld 43858
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 43841 . . 3 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
42, 3syl 17 . 2 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
51, 4eqeltrid 2843 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  cfv 6435  SAlgcsalg 43819  SalGencsalgen 43823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-int 4882  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-iota 6393  df-fun 6437  df-fv 6443  df-salg 43820  df-salgen 43824
This theorem is referenced by:  bor1sal  43864  cnfsmf  44243  incsmf  44245  bormflebmf  44256  decsmf  44269  smf2id  44302  smfco  44303
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