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Theorem salgencld 45065
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 45048 . . 3 (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
42, 3syl 17 . 2 (πœ‘ β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
51, 4eqeltrid 2838 1 (πœ‘ β†’ 𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  β€˜cfv 6544  SAlgcsalg 45024  SalGencsalgen 45028
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-salg 45025  df-salgen 45029
This theorem is referenced by:  bor1sal  45071  cnfsmf  45456  incsmf  45458  bormflebmf  45469  decsmf  45483  smf2id  45517  smfco  45518
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