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Theorem isufil 20272
Description: The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
Assertion
Ref Expression
isufil  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
Distinct variable groups:    x, F    x, X

Proof of Theorem isufil
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ufil 20270 . 2  |-  UFil  =  ( y  e.  _V  |->  { z  e.  ( Fil `  y )  |  A. x  e. 
~P  y ( x  e.  z  \/  (
y  \  x )  e.  z ) } )
2 pweq 4019 . . . 4  |-  ( y  =  X  ->  ~P y  =  ~P X
)
32adantr 465 . . 3  |-  ( ( y  =  X  /\  z  =  F )  ->  ~P y  =  ~P X )
4 eleq2 2540 . . . . 5  |-  ( z  =  F  ->  (
x  e.  z  <->  x  e.  F ) )
54adantl 466 . . . 4  |-  ( ( y  =  X  /\  z  =  F )  ->  ( x  e.  z  <-> 
x  e.  F ) )
6 difeq1 3620 . . . . 5  |-  ( y  =  X  ->  (
y  \  x )  =  ( X  \  x ) )
7 eleq12 2543 . . . . 5  |-  ( ( ( y  \  x
)  =  ( X 
\  x )  /\  z  =  F )  ->  ( ( y  \  x )  e.  z  <-> 
( X  \  x
)  e.  F ) )
86, 7sylan 471 . . . 4  |-  ( ( y  =  X  /\  z  =  F )  ->  ( ( y  \  x )  e.  z  <-> 
( X  \  x
)  e.  F ) )
95, 8orbi12d 709 . . 3  |-  ( ( y  =  X  /\  z  =  F )  ->  ( ( x  e.  z  \/  ( y 
\  x )  e.  z )  <->  ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
103, 9raleqbidv 3077 . 2  |-  ( ( y  =  X  /\  z  =  F )  ->  ( A. x  e. 
~P  y ( x  e.  z  \/  (
y  \  x )  e.  z )  <->  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
11 fveq2 5872 . 2  |-  ( y  =  X  ->  ( Fil `  y )  =  ( Fil `  X
) )
12 fvex 5882 . 2  |-  ( Fil `  y )  e.  _V
13 elfvdm 5898 . 2  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  dom  Fil )
141, 10, 11, 12, 13elmptrab2 20197 1  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    \ cdif 3478   ~Pcpw 4016   dom cdm 5005   ` cfv 5594   Filcfil 20214   UFilcufil 20268
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
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-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ufil 20270
This theorem is referenced by:  ufilfil  20273  ufilss  20274  isufil2  20277  trufil  20279  fixufil  20291  fin1aufil  20301
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