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Theorem uffixfr 19496
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 19477 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
3 filtop 19428 . . . . . . . 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 19435 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
7 intssuni 4150 . . . . . . . . 9  |-  ( F  =/=  (/)  ->  |^| F  C_  U. F )
82, 6, 73syl 20 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  U. F )
9 filunibas 19454 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
102, 9syl 16 . . . . . . . 8  |-  ( F  e.  ( UFil `  X
)  ->  U. F  =  X )
118, 10sseqtrd 3392 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  |^| F  C_  X )
1211sselda 3356 . . . . . 6  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  A  e.  X )
13 uffix 19494 . . . . . 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 19451 . . . . 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 2517 . . 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 19424 . . . . 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 4136 . . . . . 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 3430 . . . 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 19480 . . 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 1218 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  F  =  { x  e.  ~P X  |  A  e.  x } )
30 eqimss 3408 . . . . 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 2504 . . . . . 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 2504 . . . . . 6  |-  ( x  =  X  ->  ( A  e.  x  <->  A  e.  X ) )
3837elrab 3117 . . . . 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 4136 . . . 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 828 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 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   {crab 2719    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   {csn 3877   U.cuni 4091   |^|cint 4128   ` cfv 5418  (class class class)co 6091   fBascfbas 17804   filGencfg 17805   Filcfil 19418   UFilcufil 19472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-fbas 17814  df-fg 17815  df-fil 19419  df-ufil 19474
This theorem is referenced by:  uffix2  19497  uffixsn  19498
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