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Theorem dmsigagen 26723
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 3073 . . . . . . 7  |-  j  e. 
_V
21uniex 6478 . . . . . 6  |-  U. j  e.  _V
3 pwsiga 26709 . . . . . 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 4623 . . . . 5  |-  j  C_  ~P U. j
6 sseq2 3478 . . . . . 6  |-  ( s  =  ~P U. j  ->  ( j  C_  s  <->  j 
C_  ~P U. j ) )
76rspcev 3171 . . . . 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 672 . . . 4  |-  E. s  e.  (sigAlgebra `  U. j ) j  C_  s
9 rabn0 3757 . . . 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 4548 . . 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 26718 . 2  |- sigaGen  =  ( j  e.  _V  |->  |^|
{ s  e.  (sigAlgebra ` 
U. j )  |  j  C_  s }
)
1412, 13dmmpti 5640 1  |-  dom sigaGen  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796   {crab 2799   _Vcvv 3070    C_ wss 3428   (/)c0 3737   ~Pcpw 3960   U.cuni 4191   |^|cint 4228   dom cdm 4940   ` cfv 5518  sigAlgebracsiga 26686  sigaGencsigagen 26717
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 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-int 4229  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fn 5521  df-fv 5526  df-siga 26687  df-sigagen 26718
This theorem is referenced by: (None)
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