MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fmufil Structured version   Unicode version

Theorem fmufil 19530
Description: An image filter of an ultrafilter is an ultrafilter. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fmufil  |-  ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( UFil `  X
) )

Proof of Theorem fmufil
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ufilfil 19475 . . . 4  |-  ( L  e.  ( UFil `  Y
)  ->  L  e.  ( Fil `  Y ) )
2 filfbas 19419 . . . 4  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  ( fBas `  Y )
)
31, 2syl 16 . . 3  |-  ( L  e.  ( UFil `  Y
)  ->  L  e.  ( fBas `  Y )
)
4 fmfil 19515 . . 3  |-  ( ( X  e.  A  /\  L  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( Fil `  X
) )
53, 4syl3an2 1252 . 2  |-  ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( Fil `  X
) )
6 simpl2 992 . . . . . . 7  |-  ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  /\  ( f  e.  ( Fil `  X
)  /\  ( ( X  FilMap  F ) `  L )  C_  f
) )  ->  L  e.  ( UFil `  Y
) )
76, 1, 23syl 20 . . . . . 6  |-  ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  /\  ( f  e.  ( Fil `  X
)  /\  ( ( X  FilMap  F ) `  L )  C_  f
) )  ->  L  e.  ( fBas `  Y
) )
8 simprl 755 . . . . . 6  |-  ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  /\  ( f  e.  ( Fil `  X
)  /\  ( ( X  FilMap  F ) `  L )  C_  f
) )  ->  f  e.  ( Fil `  X
) )
9 simpl3 993 . . . . . 6  |-  ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  /\  ( f  e.  ( Fil `  X
)  /\  ( ( X  FilMap  F ) `  L )  C_  f
) )  ->  F : Y --> X )
10 simprr 756 . . . . . 6  |-  ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  /\  ( f  e.  ( Fil `  X
)  /\  ( ( X  FilMap  F ) `  L )  C_  f
) )  ->  (
( X  FilMap  F ) `
 L )  C_  f )
117, 8, 9, 10fmfnfm 19529 . . . . 5  |-  ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  /\  ( f  e.  ( Fil `  X
)  /\  ( ( X  FilMap  F ) `  L )  C_  f
) )  ->  E. g  e.  ( Fil `  Y
) ( L  C_  g  /\  f  =  ( ( X  FilMap  F ) `
 g ) ) )
126adantr 465 . . . . . . . 8  |-  ( ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  /\  (
f  e.  ( Fil `  X )  /\  (
( X  FilMap  F ) `
 L )  C_  f ) )  /\  ( g  e.  ( Fil `  Y )  /\  ( L  C_  g  /\  f  =  ( ( X  FilMap  F ) `
 g ) ) ) )  ->  L  e.  ( UFil `  Y
) )
13 simprl 755 . . . . . . . 8  |-  ( ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  /\  (
f  e.  ( Fil `  X )  /\  (
( X  FilMap  F ) `
 L )  C_  f ) )  /\  ( g  e.  ( Fil `  Y )  /\  ( L  C_  g  /\  f  =  ( ( X  FilMap  F ) `
 g ) ) ) )  ->  g  e.  ( Fil `  Y
) )
14 simprrl 763 . . . . . . . 8  |-  ( ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  /\  (
f  e.  ( Fil `  X )  /\  (
( X  FilMap  F ) `
 L )  C_  f ) )  /\  ( g  e.  ( Fil `  Y )  /\  ( L  C_  g  /\  f  =  ( ( X  FilMap  F ) `
 g ) ) ) )  ->  L  C_  g )
15 ufilmax 19478 . . . . . . . 8  |-  ( ( L  e.  ( UFil `  Y )  /\  g  e.  ( Fil `  Y
)  /\  L  C_  g
)  ->  L  =  g )
1612, 13, 14, 15syl3anc 1218 . . . . . . 7  |-  ( ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  /\  (
f  e.  ( Fil `  X )  /\  (
( X  FilMap  F ) `
 L )  C_  f ) )  /\  ( g  e.  ( Fil `  Y )  /\  ( L  C_  g  /\  f  =  ( ( X  FilMap  F ) `
 g ) ) ) )  ->  L  =  g )
1716fveq2d 5693 . . . . . 6  |-  ( ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  /\  (
f  e.  ( Fil `  X )  /\  (
( X  FilMap  F ) `
 L )  C_  f ) )  /\  ( g  e.  ( Fil `  Y )  /\  ( L  C_  g  /\  f  =  ( ( X  FilMap  F ) `
 g ) ) ) )  ->  (
( X  FilMap  F ) `
 L )  =  ( ( X  FilMap  F ) `  g ) )
18 simprrr 764 . . . . . 6  |-  ( ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  /\  (
f  e.  ( Fil `  X )  /\  (
( X  FilMap  F ) `
 L )  C_  f ) )  /\  ( g  e.  ( Fil `  Y )  /\  ( L  C_  g  /\  f  =  ( ( X  FilMap  F ) `
 g ) ) ) )  ->  f  =  ( ( X 
FilMap  F ) `  g
) )
1917, 18eqtr4d 2476 . . . . 5  |-  ( ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  /\  (
f  e.  ( Fil `  X )  /\  (
( X  FilMap  F ) `
 L )  C_  f ) )  /\  ( g  e.  ( Fil `  Y )  /\  ( L  C_  g  /\  f  =  ( ( X  FilMap  F ) `
 g ) ) ) )  ->  (
( X  FilMap  F ) `
 L )  =  f )
2011, 19rexlimddv 2843 . . . 4  |-  ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  /\  ( f  e.  ( Fil `  X
)  /\  ( ( X  FilMap  F ) `  L )  C_  f
) )  ->  (
( X  FilMap  F ) `
 L )  =  f )
2120expr 615 . . 3  |-  ( ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  /\  f  e.  ( Fil `  X ) )  ->  ( (
( X  FilMap  F ) `
 L )  C_  f  ->  ( ( X 
FilMap  F ) `  L
)  =  f ) )
2221ralrimiva 2797 . 2  |-  ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  ->  A. f  e.  ( Fil `  X ) ( ( ( X  FilMap  F ) `  L ) 
C_  f  ->  (
( X  FilMap  F ) `
 L )  =  f ) )
23 isufil2 19479 . 2  |-  ( ( ( X  FilMap  F ) `
 L )  e.  ( UFil `  X
)  <->  ( ( ( X  FilMap  F ) `  L )  e.  ( Fil `  X )  /\  A. f  e.  ( Fil `  X
) ( ( ( X  FilMap  F ) `  L )  C_  f  ->  ( ( X  FilMap  F ) `  L )  =  f ) ) )
245, 22, 23sylanbrc 664 1  |-  ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( UFil `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2713    C_ wss 3326   -->wf 5412   ` cfv 5416  (class class class)co 6089   fBascfbas 17802   Filcfil 19416   UFilcufil 19470    FilMap cfm 19504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-fin 7312  df-fi 7659  df-fbas 17812  df-fg 17813  df-fil 19417  df-ufil 19472  df-fm 19509
This theorem is referenced by:  ufldom  19533  uffcfflf  19610
  Copyright terms: Public domain W3C validator