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Theorem uffix2 20298
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 20278 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
2 filn0 20236 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
3 intssuni 4294 . . . . . . . 8  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
41, 2, 33syl 20 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
5 filunibas 20255 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
61, 5syl 16 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
74, 6sseqtrd 3525 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
87sseld 3488 . . . . 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 20297 . . . . 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 1701 . 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 3780 . 2  |-  ( |^| F  =/=  (/)  <->  E. x  x  e. 
|^| F )
15 df-rex 2799 . 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 1383   E.wex 1599    e. wcel 1804    =/= wne 2638   E.wrex 2794   {crab 2797    C_ wss 3461   (/)c0 3770   ~Pcpw 3997   U.cuni 4234   |^|cint 4271   ` cfv 5578   Filcfil 20219   UFilcufil 20273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-int 4272  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-fbas 18290  df-fg 18291  df-fil 20220  df-ufil 20275
This theorem is referenced by:  uffinfix  20301
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