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Theorem salgencld 45055
Description: SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
salgencld.1 (πœ‘ β†’ 𝑋 ∈ 𝑉)
salgencld.2 𝑆 = (SalGenβ€˜π‘‹)
Assertion
Ref Expression
salgencld (πœ‘ β†’ 𝑆 ∈ SAlg)

Proof of Theorem salgencld
StepHypRef Expression
1 salgencld.2 . 2 𝑆 = (SalGenβ€˜π‘‹)
2 salgencld.1 . . 3 (πœ‘ β†’ 𝑋 ∈ 𝑉)
3 salgencl 45038 . . 3 (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
42, 3syl 17 . 2 (πœ‘ β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
51, 4eqeltrid 2837 1 (πœ‘ β†’ 𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  β€˜cfv 6543  SAlgcsalg 45014  SalGencsalgen 45018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-salg 45015  df-salgen 45019
This theorem is referenced by:  bor1sal  45061  cnfsmf  45446  incsmf  45448  bormflebmf  45459  decsmf  45473  smf2id  45507  smfco  45508
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