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Theorem salgencld 45363
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 45346 . . 3 (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
42, 3syl 17 . 2 (πœ‘ β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
51, 4eqeltrid 2835 1 (πœ‘ β†’ 𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1539   ∈ wcel 2104  β€˜cfv 6542  SAlgcsalg 45322  SalGencsalgen 45326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-salg 45323  df-salgen 45327
This theorem is referenced by:  bor1sal  45369  cnfsmf  45754  incsmf  45756  bormflebmf  45767  decsmf  45781  smf2id  45815  smfco  45816
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