Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sgsiga Structured version   Visualization version   GIF version

Theorem sgsiga 31509
Description: A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypothesis
Ref Expression
sgsiga.1 (𝜑𝐴𝑉)
Assertion
Ref Expression
sgsiga (𝜑 → (sigaGen‘𝐴) ∈ ran sigAlgebra)

Proof of Theorem sgsiga
StepHypRef Expression
1 sgsiga.1 . 2 (𝜑𝐴𝑉)
2 sigagensiga 31508 . 2 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
3 elrnsiga 31493 . 2 ((sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴) → (sigaGen‘𝐴) ∈ ran sigAlgebra)
41, 2, 33syl 18 1 (𝜑 → (sigaGen‘𝐴) ∈ ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112   cuni 4803  ran crn 5524  cfv 6328  sigAlgebracsiga 31475  sigaGencsigagen 31505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fv 6336  df-siga 31476  df-sigagen 31506
This theorem is referenced by:  elsigagen2  31515  cldssbrsiga  31554  mbfmbfm  31624  imambfm  31628  sxbrsigalem2  31652  sxbrsiga  31656  sibf0  31700  sibff  31702  sibfinima  31705  sibfof  31706  sitgclg  31708  orvcval4  31826  orvcoel  31827  orvccel  31828
  Copyright terms: Public domain W3C validator