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Theorem isufl 17866
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 5668 . . 3  |-  ( x  =  X  ->  ( Fil `  x )  =  ( Fil `  X
) )
2 fveq2 5668 . . . 4  |-  ( x  =  X  ->  ( UFil `  x )  =  ( UFil `  X
) )
32rexeqdv 2854 . . 3  |-  ( x  =  X  ->  ( E. g  e.  ( UFil `  x ) f 
C_  g  <->  E. g  e.  ( UFil `  X
) f  C_  g
) )
41, 3raleqbidv 2859 . 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 17855 . 2  |- UFL  =  {
x  |  A. f  e.  ( Fil `  x
) E. g  e.  ( UFil `  x
) f  C_  g }
64, 5elab2g 3027 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 1649    e. wcel 1717   A.wral 2649   E.wrex 2650    C_ wss 3263   ` cfv 5394   Filcfil 17798   UFilcufil 17852  UFLcufl 17853
This theorem is referenced by:  ufli  17867  numufl  17868  ssufl  17871  ufldom  17915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ufl 17855
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