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Theorem dmsigagen 28974
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 3083 . . . . . . 7  |-  j  e. 
_V
21uniex 6601 . . . . . 6  |-  U. j  e.  _V
3 pwsiga 28960 . . . . . 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 4652 . . . . 5  |-  j  C_  ~P U. j
6 sseq2 3486 . . . . . 6  |-  ( s  =  ~P U. j  ->  ( j  C_  s  <->  j 
C_  ~P U. j ) )
76rspcev 3182 . . . . 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 676 . . . 4  |-  E. s  e.  (sigAlgebra `  U. j ) j  C_  s
9 rabn0 3782 . . . 4  |-  ( { s  e.  (sigAlgebra `  U. j )  |  j 
C_  s }  =/=  (/)  <->  E. s  e.  (sigAlgebra `  U. j ) j  C_  s )
108, 9mpbir 212 . . 3  |-  { s  e.  (sigAlgebra `  U. j )  |  j  C_  s }  =/=  (/)
11 intex 4580 . . 3  |-  ( { s  e.  (sigAlgebra `  U. j )  |  j 
C_  s }  =/=  (/)  <->  |^|
{ s  e.  (sigAlgebra ` 
U. j )  |  j  C_  s }  e.  _V )
1210, 11mpbi 211 . 2  |-  |^| { s  e.  (sigAlgebra `  U. j )  |  j  C_  s }  e.  _V
13 df-sigagen 28969 . 2  |- sigaGen  =  ( j  e.  _V  |->  |^|
{ s  e.  (sigAlgebra ` 
U. j )  |  j  C_  s }
)
1412, 13dmmpti 5725 1  |-  dom sigaGen  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1872    =/= wne 2614   E.wrex 2772   {crab 2775   _Vcvv 3080    C_ wss 3436   (/)c0 3761   ~Pcpw 3981   U.cuni 4219   |^|cint 4255   dom cdm 4853   ` cfv 5601  sigAlgebracsiga 28937  sigaGencsigagen 28968
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 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
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 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-int 4256  df-br 4424  df-opab 4483  df-mpt 4484  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-fn 5604  df-fv 5609  df-siga 28938  df-sigagen 28969
This theorem is referenced by: (None)
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