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Theorem uffix2 19630
Description: A classification of fixed ultrafilters. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
uffix2  |-  ( F  e.  ( UFil `  X
)  ->  ( |^| F  =/=  (/)  <->  E. x  e.  X  F  =  { y  e.  ~P X  |  x  e.  y } ) )
Distinct variable groups:    x, y, F    x, X, y

Proof of Theorem uffix2
StepHypRef Expression
1 ufilfil 19610 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
2 filn0 19568 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
3 intssuni 4259 . . . . . . . 8  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
41, 2, 33syl 20 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
5 filunibas 19587 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
61, 5syl 16 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
74, 6sseqtrd 3501 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
87sseld 3464 . . . . 5  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  |^| F  ->  x  e.  X ) )
98pm4.71rd 635 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  |^| F  <->  ( x  e.  X  /\  x  e.  |^| F ) ) )
10 uffixfr 19629 . . . . 5  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  |^| F  <->  F  =  { y  e.  ~P X  |  x  e.  y } ) )
1110anbi2d 703 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  ( (
x  e.  X  /\  x  e.  |^| F )  <-> 
( x  e.  X  /\  F  =  {
y  e.  ~P X  |  x  e.  y } ) ) )
129, 11bitrd 253 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  |^| F  <->  ( x  e.  X  /\  F  =  { y  e.  ~P X  |  x  e.  y } ) ) )
1312exbidv 1681 . 2  |-  ( F  e.  ( UFil `  X
)  ->  ( E. x  x  e.  |^| F  <->  E. x ( x  e.  X  /\  F  =  { y  e.  ~P X  |  x  e.  y } ) ) )
14 n0 3755 . 2  |-  ( |^| F  =/=  (/)  <->  E. x  x  e. 
|^| F )
15 df-rex 2805 . 2  |-  ( E. x  e.  X  F  =  { y  e.  ~P X  |  x  e.  y }  <->  E. x ( x  e.  X  /\  F  =  { y  e.  ~P X  |  x  e.  y } ) )
1613, 14, 153bitr4g 288 1  |-  ( F  e.  ( UFil `  X
)  ->  ( |^| F  =/=  (/)  <->  E. x  e.  X  F  =  { y  e.  ~P X  |  x  e.  y } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   E.wrex 2800   {crab 2803    C_ wss 3437   (/)c0 3746   ~Pcpw 3969   U.cuni 4200   |^|cint 4237   ` cfv 5527   Filcfil 19551   UFilcufil 19605
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-int 4238  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-fbas 17940  df-fg 17941  df-fil 19552  df-ufil 19607
This theorem is referenced by:  uffinfix  19633
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