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Theorem sssigagen2 33935
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 483 . 2 ((𝐴𝑉𝐵𝐴) → 𝐵𝐴)
2 sssigagen 33934 . . 3 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
32adantr 479 . 2 ((𝐴𝑉𝐵𝐴) → 𝐴 ⊆ (sigaGen‘𝐴))
41, 3sstrd 3989 1 ((𝐴𝑉𝐵𝐴) → 𝐵 ⊆ (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  wss 3946  cfv 6553  sigaGencsigagen 33927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-iota 6505  df-fun 6555  df-fv 6561  df-siga 33898  df-sigagen 33928
This theorem is referenced by:  sxbrsigalem5  34078
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