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Theorem isufl 17608
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 5525 . . 3  |-  ( x  =  X  ->  ( Fil `  x )  =  ( Fil `  X
) )
2 fveq2 5525 . . . 4  |-  ( x  =  X  ->  ( UFil `  x )  =  ( UFil `  X
) )
32rexeqdv 2743 . . 3  |-  ( x  =  X  ->  ( E. g  e.  ( UFil `  x ) f 
C_  g  <->  E. g  e.  ( UFil `  X
) f  C_  g
) )
41, 3raleqbidv 2748 . 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 17597 . 2  |- UFL  =  {
x  |  A. f  e.  ( Fil `  x
) E. g  e.  ( UFil `  x
) f  C_  g }
64, 5elab2g 2916 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 176    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   ` cfv 5255   Filcfil 17540   UFilcufil 17594  UFLcufl 17595
This theorem is referenced by:  ufli  17609  numufl  17610  ssufl  17613  ufldom  17657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ufl 17597
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