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Theorem sssigagen2 31083
Description: A subset of the generating set is also a subset of the generated sigma-algebra. (Contributed by Thierry Arnoux, 22-Sep-2017.)
Assertion
Ref Expression
sssigagen2 ((𝐴𝑉𝐵𝐴) → 𝐵 ⊆ (sigaGen‘𝐴))

Proof of Theorem sssigagen2
StepHypRef Expression
1 simpr 477 . 2 ((𝐴𝑉𝐵𝐴) → 𝐵𝐴)
2 sssigagen 31082 . . 3 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
32adantr 473 . 2 ((𝐴𝑉𝐵𝐴) → 𝐴 ⊆ (sigaGen‘𝐴))
41, 3sstrd 3863 1 ((𝐴𝑉𝐵𝐴) → 𝐵 ⊆ (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wcel 2051  wss 3824  cfv 6186  sigaGencsigagen 31075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-int 4747  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-iota 6150  df-fun 6188  df-fv 6194  df-siga 31045  df-sigagen 31076
This theorem is referenced by:  sxbrsigalem5  31224
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