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Theorem unisg 30515
 Description: The sigma-algebra generated by a collection 𝐴 is a sigma-algebra on ∪ 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
unisg (𝐴𝑉 (sigaGen‘𝐴) = 𝐴)

Proof of Theorem unisg
StepHypRef Expression
1 sigagensiga 30513 . . . 4 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
2 issgon 30495 . . . 4 ((sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴) ↔ ((sigaGen‘𝐴) ∈ ran sigAlgebra ∧ 𝐴 = (sigaGen‘𝐴)))
31, 2sylib 208 . . 3 (𝐴𝑉 → ((sigaGen‘𝐴) ∈ ran sigAlgebra ∧ 𝐴 = (sigaGen‘𝐴)))
43simprd 482 . 2 (𝐴𝑉 𝐴 = (sigaGen‘𝐴))
54eqcomd 2766 1 (𝐴𝑉 (sigaGen‘𝐴) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∪ cuni 4588  ran crn 5267  ‘cfv 6049  sigAlgebracsiga 30479  sigaGencsigagen 30510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-siga 30480  df-sigagen 30511 This theorem is referenced by:  unibrsiga  30558  sxsigon  30564  imambfm  30633  cnmbfm  30634  sibf0  30705  sibff  30707  sibfof  30711  sitgclg  30713  orvcval4  30831
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