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Theorem sigagenss 32812
Description: The generated sigma-algebra is a subset of all sigma-algebras containing the generating set, i.e. the generated sigma-algebra is the smallest sigma-algebra containing the generating set, here 𝐴. (Contributed by Thierry Arnoux, 4-Jun-2017.)
Assertion
Ref Expression
sigagenss ((𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∧ 𝐴 βŠ† 𝑆) β†’ (sigaGenβ€˜π΄) βŠ† 𝑆)

Proof of Theorem sigagenss
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ssexg 5284 . . . 4 ((𝐴 βŠ† 𝑆 ∧ 𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝐴)) β†’ 𝐴 ∈ V)
21ancoms 460 . . 3 ((𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∧ 𝐴 βŠ† 𝑆) β†’ 𝐴 ∈ V)
3 sigagenval 32803 . . 3 (𝐴 ∈ V β†’ (sigaGenβ€˜π΄) = ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
42, 3syl 17 . 2 ((𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∧ 𝐴 βŠ† 𝑆) β†’ (sigaGenβ€˜π΄) = ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
5 sseq2 3974 . . 3 (𝑠 = 𝑆 β†’ (𝐴 βŠ† 𝑠 ↔ 𝐴 βŠ† 𝑆))
65intminss 4939 . 2 ((𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∧ 𝐴 βŠ† 𝑆) β†’ ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠} βŠ† 𝑆)
74, 6eqsstrd 3986 1 ((𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∧ 𝐴 βŠ† 𝑆) β†’ (sigaGenβ€˜π΄) βŠ† 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3447   βŠ† wss 3914  βˆͺ cuni 4869  βˆ© cint 4911  β€˜cfv 6500  sigAlgebracsiga 32771  sigaGencsigagen 32801
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 2704  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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  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-siga 32772  df-sigagen 32802
This theorem is referenced by:  sigagenss2  32813  sigagenid  32814  imambfm  32926
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