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Theorem salgencld 44680
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 44663 . . 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 1542   ∈ wcel 2107  β€˜cfv 6500  SAlgcsalg 44639  SalGencsalgen 44643
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 2703  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 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 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-salg 44640  df-salgen 44644
This theorem is referenced by:  bor1sal  44686  cnfsmf  45071  incsmf  45073  bormflebmf  45084  decsmf  45098  smf2id  45132  smfco  45133
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