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Theorem sigagenval 29011
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 29010 . . 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 4220 . . . . . 6  |-  ( x  =  A  ->  U. x  =  U. A )
43fveq2d 5892 . . . . 5  |-  ( x  =  A  ->  (sigAlgebra ` 
U. x )  =  (sigAlgebra `  U. A ) )
5 sseq1 3465 . . . . 5  |-  ( x  =  A  ->  (
x  C_  s  <->  A  C_  s
) )
64, 5rabeqbidv 3052 . . . 4  |-  ( x  =  A  ->  { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
76inteqd 4253 . . 3  |-  ( x  =  A  ->  |^| { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
87adantl 472 . 2  |-  ( ( A  e.  V  /\  x  =  A )  ->  |^| { s  e.  (sigAlgebra `  U. x )  |  x  C_  s }  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
9 elex 3066 . 2  |-  ( A  e.  V  ->  A  e.  _V )
10 uniexg 6615 . . . . . . 7  |-  ( A  e.  V  ->  U. A  e.  _V )
11 pwsiga 29001 . . . . . . 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 4645 . . . . . 6  |-  A  C_  ~P U. A
1412, 13jctir 545 . . . . 5  |-  ( A  e.  V  ->  ( ~P U. A  e.  (sigAlgebra ` 
U. A )  /\  A  C_  ~P U. A
) )
15 sseq2 3466 . . . . . 6  |-  ( s  =  ~P U. A  ->  ( A  C_  s  <->  A 
C_  ~P U. A ) )
1615elrab 3208 . . . . 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 217 . . . 4  |-  ( A  e.  V  ->  ~P U. A  e.  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
18 ne0i 3749 . . . 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 4573 . . 3  |-  ( { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  =/=  (/)  <->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  _V )
2119, 20sylib 201 . 2  |-  ( A  e.  V  ->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  _V )
222, 8, 9, 21fvmptd 5977 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 375    = wceq 1455    e. wcel 1898    =/= wne 2633   {crab 2753   _Vcvv 3057    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   U.cuni 4212   |^|cint 4248    |-> cmpt 4475   ` cfv 5601  sigAlgebracsiga 28978  sigaGencsigagen 29009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-int 4249  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-siga 28979  df-sigagen 29010
This theorem is referenced by:  sigagensiga  29012  sssigagen  29016  sigagenss  29020
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