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Theorem isufl 17937
Description: Define the (strong) ultrafilter lemma, parameterized over base sets. A set  X satisfies the ultrafilter lemma if every filter on  X is a subset of some ultrafilter. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
isufl  |-  ( X  e.  V  ->  ( X  e. UFL  <->  A. f  e.  ( Fil `  X ) E. g  e.  (
UFil `  X )
f  C_  g )
)
Distinct variable group:    f, g, X
Allowed substitution hints:    V( f, g)

Proof of Theorem isufl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . 3  |-  ( x  =  X  ->  ( Fil `  x )  =  ( Fil `  X
) )
2 fveq2 5720 . . . 4  |-  ( x  =  X  ->  ( UFil `  x )  =  ( UFil `  X
) )
32rexeqdv 2903 . . 3  |-  ( x  =  X  ->  ( E. g  e.  ( UFil `  x ) f 
C_  g  <->  E. g  e.  ( UFil `  X
) f  C_  g
) )
41, 3raleqbidv 2908 . 2  |-  ( x  =  X  ->  ( A. f  e.  ( Fil `  x ) E. g  e.  ( UFil `  x ) f  C_  g 
<-> 
A. f  e.  ( Fil `  X ) E. g  e.  (
UFil `  X )
f  C_  g )
)
5 df-ufl 17926 . 2  |- UFL  =  {
x  |  A. f  e.  ( Fil `  x
) E. g  e.  ( UFil `  x
) f  C_  g }
64, 5elab2g 3076 1  |-  ( X  e.  V  ->  ( X  e. UFL  <->  A. f  e.  ( Fil `  X ) E. g  e.  (
UFil `  X )
f  C_  g )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   ` cfv 5446   Filcfil 17869   UFilcufil 17923  UFLcufl 17924
This theorem is referenced by:  ufli  17938  numufl  17939  ssufl  17942  ufldom  17986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ufl 17926
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