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Theorem salgencld 45970
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 45953 . . 3 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
42, 3syl 17 . 2 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
51, 4eqeltrid 2830 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cfv 6554  SAlgcsalg 45929  SalGencsalgen 45933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562  df-salg 45930  df-salgen 45934
This theorem is referenced by:  bor1sal  45976  cnfsmf  46361  incsmf  46363  bormflebmf  46374  decsmf  46388  smf2id  46422  smfco  46423
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