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Theorem sigagenval 28914
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 28913 . . 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 4170 . . . . . 6  |-  ( x  =  A  ->  U. x  =  U. A )
43fveq2d 5829 . . . . 5  |-  ( x  =  A  ->  (sigAlgebra ` 
U. x )  =  (sigAlgebra `  U. A ) )
5 sseq1 3428 . . . . 5  |-  ( x  =  A  ->  (
x  C_  s  <->  A  C_  s
) )
64, 5rabeqbidv 3017 . . . 4  |-  ( x  =  A  ->  { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
76inteqd 4203 . . 3  |-  ( x  =  A  ->  |^| { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
87adantl 467 . 2  |-  ( ( A  e.  V  /\  x  =  A )  ->  |^| { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
9 elex 3031 . 2  |-  ( A  e.  V  ->  A  e.  _V )
10 uniexg 6546 . . . . . . 7  |-  ( A  e.  V  ->  U. A  e.  _V )
11 pwsiga 28904 . . . . . . 7  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  (sigAlgebra ` 
U. A ) )
1210, 11syl 17 . . . . . 6  |-  ( A  e.  V  ->  ~P U. A  e.  (sigAlgebra `  U. A ) )
13 pwuni 4595 . . . . . 6  |-  A  C_  ~P U. A
1412, 13jctir 540 . . . . 5  |-  ( A  e.  V  ->  ( ~P U. A  e.  (sigAlgebra ` 
U. A )  /\  A  C_  ~P U. A
) )
15 sseq2 3429 . . . . . 6  |-  ( s  =  ~P U. A  ->  ( A  C_  s  <->  A 
C_  ~P U. A ) )
1615elrab 3171 . . . . 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 215 . . . 4  |-  ( A  e.  V  ->  ~P U. A  e.  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
18 ne0i 3710 . . . 4  |-  ( ~P
U. A  e.  {
s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  ->  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  =/=  (/) )
1917, 18syl 17 . . 3  |-  ( A  e.  V  ->  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  =/=  (/) )
20 intex 4523 . . 3  |-  ( { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  =/=  (/)  <->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  _V )
2119, 20sylib 199 . 2  |-  ( A  e.  V  ->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  _V )
222, 8, 9, 21fvmptd 5914 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 370    = wceq 1437    e. wcel 1872    =/= wne 2599   {crab 2718   _Vcvv 3022    C_ wss 3379   (/)c0 3704   ~Pcpw 3924   U.cuni 4162   |^|cint 4198    |-> cmpt 4425   ` cfv 5544  sigAlgebracsiga 28881  sigaGencsigagen 28912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-int 4199  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-iota 5508  df-fun 5546  df-fv 5552  df-siga 28882  df-sigagen 28913
This theorem is referenced by:  sigagensiga  28915  sssigagen  28919  sigagenss  28923
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