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Theorem sigagenval 28370
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 28369 . . 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 4243 . . . . . 6  |-  ( x  =  A  ->  U. x  =  U. A )
43fveq2d 5852 . . . . 5  |-  ( x  =  A  ->  (sigAlgebra ` 
U. x )  =  (sigAlgebra `  U. A ) )
5 sseq1 3510 . . . . 5  |-  ( x  =  A  ->  (
x  C_  s  <->  A  C_  s
) )
64, 5rabeqbidv 3101 . . . 4  |-  ( x  =  A  ->  { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
76inteqd 4276 . . 3  |-  ( x  =  A  ->  |^| { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
87adantl 464 . 2  |-  ( ( A  e.  V  /\  x  =  A )  ->  |^| { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
9 elex 3115 . 2  |-  ( A  e.  V  ->  A  e.  _V )
10 uniexg 6570 . . . . . . 7  |-  ( A  e.  V  ->  U. A  e.  _V )
11 pwsiga 28360 . . . . . . 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 4668 . . . . . 6  |-  A  C_  ~P U. A
1412, 13jctir 536 . . . . 5  |-  ( A  e.  V  ->  ( ~P U. A  e.  (sigAlgebra ` 
U. A )  /\  A  C_  ~P U. A
) )
15 sseq2 3511 . . . . . 6  |-  ( s  =  ~P U. A  ->  ( A  C_  s  <->  A 
C_  ~P U. A ) )
1615elrab 3254 . . . . 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 3789 . . . 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 4593 . . 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 5936 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   U.cuni 4235   |^|cint 4271    |-> cmpt 4497   ` cfv 5570  sigAlgebracsiga 28337  sigaGencsigagen 28368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-siga 28338  df-sigagen 28369
This theorem is referenced by:  sigagensiga  28371  sssigagen  28375  sigagenss  28379
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