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Theorem uffixfr 20152
Description: An ultrafilter is either fixed or free. A fixed ultrafilter is called principal (generated by a single element  A), and a free ultrafilter is called nonprincipal (having empty intersection). Note that examples of free ultrafilters cannot be defined in ZFC without some form of global choice. (Contributed by Jeff Hankins, 4-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
uffixfr  |-  ( F  e.  ( UFil `  X
)  ->  ( A  e.  |^| F  <->  F  =  { x  e.  ~P X  |  A  e.  x } ) )
Distinct variable groups:    x, A    x, F    x, X

Proof of Theorem uffixfr
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  e.  ( UFil `  X ) )
2 ufilfil 20133 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
3 filtop 20084 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
42, 3syl 16 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  X  e.  F )
54adantr 465 . . . . . 6  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  X  e.  F )
6 filn0 20091 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
7 intssuni 4297 . . . . . . . . 9  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
82, 6, 73syl 20 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
9 filunibas 20110 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
102, 9syl 16 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
118, 10sseqtrd 3533 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
1211sselda 3497 . . . . . 6  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  X )
13 uffix 20150 . . . . . 6  |-  ( ( X  e.  F  /\  A  e.  X )  ->  ( { { A } }  e.  ( fBas `  X )  /\  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) ) )
145, 12, 13syl2anc 661 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  -> 
( { { A } }  e.  ( fBas `  X )  /\  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) ) )
1514simprd 463 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } } ) )
1614simpld 459 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { { A } }  e.  ( fBas `  X
) )
17 fgcl 20107 . . . . 5  |-  ( { { A } }  e.  ( fBas `  X
)  ->  ( X filGen { { A } } )  e.  ( Fil `  X ) )
1816, 17syl 16 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  -> 
( X filGen { { A } } )  e.  ( Fil `  X
) )
1915, 18eqeltrd 2548 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { x  e.  ~P X  |  A  e.  x }  e.  ( Fil `  X ) )
202adantr 465 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  e.  ( Fil `  X ) )
21 filsspw 20080 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  C_  ~P X )
2220, 21syl 16 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  C_  ~P X )
23 elintg 4283 . . . . . 6  |-  ( A  e.  |^| F  ->  ( A  e.  |^| F  <->  A. x  e.  F  A  e.  x ) )
2423ibi 241 . . . . 5  |-  ( A  e.  |^| F  ->  A. x  e.  F  A  e.  x )
2524adantl 466 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A. x  e.  F  A  e.  x )
26 ssrab 3571 . . . 4  |-  ( F 
C_  { x  e. 
~P X  |  A  e.  x }  <->  ( F  C_ 
~P X  /\  A. x  e.  F  A  e.  x ) )
2722, 25, 26sylanbrc 664 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  C_  { x  e. 
~P X  |  A  e.  x } )
28 ufilmax 20136 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  {
x  e.  ~P X  |  A  e.  x }  e.  ( Fil `  X )  /\  F  C_ 
{ x  e.  ~P X  |  A  e.  x } )  ->  F  =  { x  e.  ~P X  |  A  e.  x } )
291, 19, 27, 28syl3anc 1223 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  =  { x  e.  ~P X  |  A  e.  x } )
30 eqimss 3549 . . . . 5  |-  ( F  =  { x  e. 
~P X  |  A  e.  x }  ->  F  C_ 
{ x  e.  ~P X  |  A  e.  x } )
3130adantl 466 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  F  C_ 
{ x  e.  ~P X  |  A  e.  x } )
3226simprbi 464 . . . 4  |-  ( F 
C_  { x  e. 
~P X  |  A  e.  x }  ->  A. x  e.  F  A  e.  x )
3331, 32syl 16 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  A. x  e.  F  A  e.  x )
34 eleq2 2533 . . . . . 6  |-  ( F  =  { x  e. 
~P X  |  A  e.  x }  ->  ( X  e.  F  <->  X  e.  { x  e.  ~P X  |  A  e.  x } ) )
3534biimpac 486 . . . . 5  |-  ( ( X  e.  F  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  X  e.  { x  e.  ~P X  |  A  e.  x } )
364, 35sylan 471 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  X  e.  { x  e.  ~P X  |  A  e.  x } )
37 eleq2 2533 . . . . . 6  |-  ( x  =  X  ->  ( A  e.  x  <->  A  e.  X ) )
3837elrab 3254 . . . . 5  |-  ( X  e.  { x  e. 
~P X  |  A  e.  x }  <->  ( X  e.  ~P X  /\  A  e.  X ) )
3938simprbi 464 . . . 4  |-  ( X  e.  { x  e. 
~P X  |  A  e.  x }  ->  A  e.  X )
40 elintg 4283 . . . 4  |-  ( A  e.  X  ->  ( A  e.  |^| F  <->  A. x  e.  F  A  e.  x ) )
4136, 39, 403syl 20 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  ( A  e.  |^| F  <->  A. x  e.  F  A  e.  x ) )
4233, 41mpbird 232 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  F  =  { x  e.  ~P X  |  A  e.  x } )  ->  A  e.  |^| F )
4329, 42impbida 829 1  |-  ( F  e.  ( UFil `  X
)  ->  ( A  e.  |^| F  <->  F  =  { x  e.  ~P X  |  A  e.  x } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   {crab 2811    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   {csn 4020   U.cuni 4238   |^|cint 4275   ` cfv 5579  (class class class)co 6275   fBascfbas 18170   filGencfg 18171   Filcfil 20074   UFilcufil 20128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-fbas 18180  df-fg 18181  df-fil 20075  df-ufil 20130
This theorem is referenced by:  uffix2  20153  uffixsn  20154
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