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Theorem fmufil 20642
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 20587 . . . 4  |-  ( L  e.  ( UFil `  Y
)  ->  L  e.  ( Fil `  Y ) )
2 filfbas 20531 . . . 4  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  ( fBas `  Y )
)
31, 2syl 17 . . 3  |-  ( L  e.  ( UFil `  Y
)  ->  L  e.  ( fBas `  Y )
)
4 fmfil 20627 . . 3  |-  ( ( X  e.  A  /\  L  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( Fil `  X
) )
53, 4syl3an2 1262 . 2  |-  ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( Fil `  X
) )
6 simpl2 999 . . . . . . 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 1000 . . . . . 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 20641 . . . . 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 463 . . . . . . . 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 764 . . . . . . . 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 20590 . . . . . . . 8  |-  ( ( L  e.  ( UFil `  Y )  /\  g  e.  ( Fil `  Y
)  /\  L  C_  g
)  ->  L  =  g )
1612, 13, 14, 15syl3anc 1228 . . . . . . 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 5807 . . . . . 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 765 . . . . . 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 2444 . . . . 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 2897 . . . 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 613 . . 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 2815 . 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 20591 . 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 662 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 367    /\ w3a 972    = wceq 1403    e. wcel 1840   A.wral 2751    C_ wss 3411   -->wf 5519   ` cfv 5523  (class class class)co 6232   fBascfbas 18616   Filcfil 20528   UFilcufil 20582    FilMap cfm 20616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-en 7473  df-fin 7476  df-fi 7823  df-fbas 18626  df-fg 18627  df-fil 20529  df-ufil 20584  df-fm 20621
This theorem is referenced by:  ufldom  20645  uffcfflf  20722
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