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Theorem sigagenval 26723
Description: Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
sigagenval  |-  ( A  e.  V  ->  (sigaGen `  A )  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
Distinct variable group:    A, s
Allowed substitution hint:    V( s)

Proof of Theorem sigagenval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-sigagen 26722 . . 3  |- sigaGen  =  ( x  e.  _V  |->  |^|
{ s  e.  (sigAlgebra ` 
U. x )  |  x  C_  s }
)
21a1i 11 . 2  |-  ( A  e.  V  -> sigaGen  =  ( x  e.  _V  |->  |^|
{ s  e.  (sigAlgebra ` 
U. x )  |  x  C_  s }
) )
3 unieq 4202 . . . . . 6  |-  ( x  =  A  ->  U. x  =  U. A )
43fveq2d 5798 . . . . 5  |-  ( x  =  A  ->  (sigAlgebra ` 
U. x )  =  (sigAlgebra `  U. A ) )
5 sseq1 3480 . . . . 5  |-  ( x  =  A  ->  (
x  C_  s  <->  A  C_  s
) )
64, 5rabeqbidv 3067 . . . 4  |-  ( x  =  A  ->  { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
76inteqd 4236 . . 3  |-  ( x  =  A  ->  |^| { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
87adantl 466 . 2  |-  ( ( A  e.  V  /\  x  =  A )  ->  |^| { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
9 elex 3081 . 2  |-  ( A  e.  V  ->  A  e.  _V )
10 uniexg 6482 . . . . . . 7  |-  ( A  e.  V  ->  U. A  e.  _V )
11 pwsiga 26713 . . . . . . 7  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  (sigAlgebra ` 
U. A ) )
1210, 11syl 16 . . . . . 6  |-  ( A  e.  V  ->  ~P U. A  e.  (sigAlgebra `  U. A ) )
13 pwuni 4626 . . . . . 6  |-  A  C_  ~P U. A
1412, 13jctir 538 . . . . 5  |-  ( A  e.  V  ->  ( ~P U. A  e.  (sigAlgebra ` 
U. A )  /\  A  C_  ~P U. A
) )
15 sseq2 3481 . . . . . 6  |-  ( s  =  ~P U. A  ->  ( A  C_  s  <->  A 
C_  ~P U. A ) )
1615elrab 3218 . . . . 5  |-  ( ~P
U. A  e.  {
s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  <->  ( ~P U. A  e.  (sigAlgebra `  U. A )  /\  A  C_ 
~P U. A ) )
1714, 16sylibr 212 . . . 4  |-  ( A  e.  V  ->  ~P U. A  e.  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
18 ne0i 3746 . . . 4  |-  ( ~P
U. A  e.  {
s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  ->  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  =/=  (/) )
1917, 18syl 16 . . 3  |-  ( A  e.  V  ->  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  =/=  (/) )
20 intex 4551 . . 3  |-  ( { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  =/=  (/)  <->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  _V )
2119, 20sylib 196 . 2  |-  ( A  e.  V  ->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  _V )
222, 8, 9, 21fvmptd 5883 1  |-  ( A  e.  V  ->  (sigaGen `  A )  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   {crab 2800   _Vcvv 3072    C_ wss 3431   (/)c0 3740   ~Pcpw 3963   U.cuni 4194   |^|cint 4231    |-> cmpt 4453   ` cfv 5521  sigAlgebracsiga 26690  sigaGencsigagen 26721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-int 4232  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-siga 26691  df-sigagen 26722
This theorem is referenced by:  sigagensiga  26724  sssigagen  26728  sigagenss  26732
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