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Theorem isufil 17937
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 17935 . 2  |-  UFil  =  ( y  e.  _V  |->  { z  e.  ( Fil `  y )  |  A. x  e. 
~P  y ( x  e.  z  \/  (
y  \  x )  e.  z ) } )
2 pweq 3804 . . . 4  |-  ( y  =  X  ->  ~P y  =  ~P X
)
32adantr 453 . . 3  |-  ( ( y  =  X  /\  z  =  F )  ->  ~P y  =  ~P X )
4 eleq2 2499 . . . . 5  |-  ( z  =  F  ->  (
x  e.  z  <->  x  e.  F ) )
54adantl 454 . . . 4  |-  ( ( y  =  X  /\  z  =  F )  ->  ( x  e.  z  <-> 
x  e.  F ) )
6 difeq1 3460 . . . . 5  |-  ( y  =  X  ->  (
y  \  x )  =  ( X  \  x ) )
7 eleq12 2500 . . . . 5  |-  ( ( ( y  \  x
)  =  ( X 
\  x )  /\  z  =  F )  ->  ( ( y  \  x )  e.  z  <-> 
( X  \  x
)  e.  F ) )
86, 7sylan 459 . . . 4  |-  ( ( y  =  X  /\  z  =  F )  ->  ( ( y  \  x )  e.  z  <-> 
( X  \  x
)  e.  F ) )
95, 8orbi12d 692 . . 3  |-  ( ( y  =  X  /\  z  =  F )  ->  ( ( x  e.  z  \/  ( y 
\  x )  e.  z )  <->  ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
103, 9raleqbidv 2918 . 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 5730 . 2  |-  ( y  =  X  ->  ( Fil `  y )  =  ( Fil `  X
) )
12 fvex 5744 . 2  |-  ( Fil `  y )  e.  _V
13 elfvdm 5759 . 2  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  dom  Fil )
141, 10, 11, 12, 13elmptrab2 17862 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 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    \ cdif 3319   ~Pcpw 3801   dom cdm 4880   ` cfv 5456   Filcfil 17879   UFilcufil 17933
This theorem is referenced by:  ufilfil  17938  ufilss  17939  isufil2  17942  trufil  17944  fixufil  17956  fin1aufil  17966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-ufil 17935
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