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Theorem isufil 17614
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 17612 . 2  |-  UFil  =  ( y  e.  _V  |->  { z  e.  ( Fil `  y )  |  A. x  e. 
~P  y ( x  e.  z  \/  (
y  \  x )  e.  z ) } )
2 pweq 3641 . . . 4  |-  ( y  =  X  ->  ~P y  =  ~P X
)
32adantr 451 . . 3  |-  ( ( y  =  X  /\  z  =  F )  ->  ~P y  =  ~P X )
4 eleq2 2357 . . . . 5  |-  ( z  =  F  ->  (
x  e.  z  <->  x  e.  F ) )
54adantl 452 . . . 4  |-  ( ( y  =  X  /\  z  =  F )  ->  ( x  e.  z  <-> 
x  e.  F ) )
6 difeq1 3300 . . . . 5  |-  ( y  =  X  ->  (
y  \  x )  =  ( X  \  x ) )
7 eleq12 2358 . . . . 5  |-  ( ( ( y  \  x
)  =  ( X 
\  x )  /\  z  =  F )  ->  ( ( y  \  x )  e.  z  <-> 
( X  \  x
)  e.  F ) )
86, 7sylan 457 . . . 4  |-  ( ( y  =  X  /\  z  =  F )  ->  ( ( y  \  x )  e.  z  <-> 
( X  \  x
)  e.  F ) )
95, 8orbi12d 690 . . 3  |-  ( ( y  =  X  /\  z  =  F )  ->  ( ( x  e.  z  \/  ( y 
\  x )  e.  z )  <->  ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
103, 9raleqbidv 2761 . 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 5541 . 2  |-  ( y  =  X  ->  ( Fil `  y )  =  ( Fil `  X
) )
12 fvex 5555 . 2  |-  ( Fil `  y )  e.  _V
13 elfvdm 5570 . 2  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  dom  Fil )
141, 10, 11, 12, 13elmptrab2 17539 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 set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162   ~Pcpw 3638   dom cdm 4705   ` cfv 5271   Filcfil 17556   UFilcufil 17610
This theorem is referenced by:  ufilfil  17615  ufilss  17616  isufil2  17619  trufil  17621  fixufil  17633  fin1aufil  17643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ufil 17612
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