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Theorem uffix2 20157
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 20137 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
2 filn0 20095 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
3 intssuni 4304 . . . . . . . 8  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
41, 2, 33syl 20 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
5 filunibas 20114 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
61, 5syl 16 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
74, 6sseqtrd 3540 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
87sseld 3503 . . . . 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 20156 . . . . 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 1690 . 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 3794 . 2  |-  ( |^| F  =/=  (/)  <->  E. x  x  e. 
|^| F )
15 df-rex 2820 . 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 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   U.cuni 4245   |^|cint 4282   ` cfv 5586   Filcfil 20078   UFilcufil 20132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-fbas 18184  df-fg 18185  df-fil 20079  df-ufil 20134
This theorem is referenced by:  uffinfix  20160
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