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Theorem salgencld 41358
 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 41341 . . 3 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
42, 3syl 17 . 2 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
51, 4syl5eqel 2910 1 (𝜑𝑆 ∈ SAlg)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  ‘cfv 6123  SAlgcsalg 41319  SalGencsalgen 41323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-salg 41320  df-salgen 41324 This theorem is referenced by:  bor1sal  41364  cnfsmf  41743  incsmf  41745  bormflebmf  41756  decsmf  41769  smf2id  41802  smfco  41803
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