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Theorem dmsigagen 28324
Description: A sigma algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Assertion
Ref Expression
dmsigagen  |-  dom sigaGen  =  _V

Proof of Theorem dmsigagen
Dummy variables  j 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3050 . . . . . . 7  |-  j  e. 
_V
21uniex 6513 . . . . . 6  |-  U. j  e.  _V
3 pwsiga 28310 . . . . . 6  |-  ( U. j  e.  _V  ->  ~P
U. j  e.  (sigAlgebra ` 
U. j ) )
42, 3ax-mp 5 . . . . 5  |-  ~P U. j  e.  (sigAlgebra `  U. j )
5 pwuni 4606 . . . . 5  |-  j  C_  ~P U. j
6 sseq2 3452 . . . . . 6  |-  ( s  =  ~P U. j  ->  ( j  C_  s  <->  j 
C_  ~P U. j ) )
76rspcev 3148 . . . . 5  |-  ( ( ~P U. j  e.  (sigAlgebra `  U. j )  /\  j  C_  ~P U. j )  ->  E. s  e.  (sigAlgebra `  U. j ) j  C_  s )
84, 5, 7mp2an 670 . . . 4  |-  E. s  e.  (sigAlgebra `  U. j ) j  C_  s
9 rabn0 3745 . . . 4  |-  ( { s  e.  (sigAlgebra `  U. j )  |  j 
C_  s }  =/=  (/)  <->  E. s  e.  (sigAlgebra `  U. j ) j  C_  s )
108, 9mpbir 209 . . 3  |-  { s  e.  (sigAlgebra `  U. j )  |  j  C_  s }  =/=  (/)
11 intex 4534 . . 3  |-  ( { s  e.  (sigAlgebra `  U. j )  |  j 
C_  s }  =/=  (/)  <->  |^|
{ s  e.  (sigAlgebra ` 
U. j )  |  j  C_  s }  e.  _V )
1210, 11mpbi 208 . 2  |-  |^| { s  e.  (sigAlgebra `  U. j )  |  j  C_  s }  e.  _V
13 df-sigagen 28319 . 2  |- sigaGen  =  ( j  e.  _V  |->  |^|
{ s  e.  (sigAlgebra ` 
U. j )  |  j  C_  s }
)
1412, 13dmmpti 5631 1  |-  dom sigaGen  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399    e. wcel 1836    =/= wne 2587   E.wrex 2743   {crab 2746   _Vcvv 3047    C_ wss 3402   (/)c0 3724   ~Pcpw 3940   U.cuni 4176   |^|cint 4212   dom cdm 4926   ` cfv 5509  sigAlgebracsiga 28287  sigaGencsigagen 28318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-int 4213  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-iota 5473  df-fun 5511  df-fn 5512  df-fv 5517  df-siga 28288  df-sigagen 28319
This theorem is referenced by: (None)
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