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Theorem fmufil 20188
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 20133 . . . 4  |-  ( L  e.  ( UFil `  Y
)  ->  L  e.  ( Fil `  Y ) )
2 filfbas 20077 . . . 4  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  ( fBas `  Y )
)
31, 2syl 16 . . 3  |-  ( L  e.  ( UFil `  Y
)  ->  L  e.  ( fBas `  Y )
)
4 fmfil 20173 . . 3  |-  ( ( X  e.  A  /\  L  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( Fil `  X
) )
53, 4syl3an2 1257 . 2  |-  ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( Fil `  X
) )
6 simpl2 995 . . . . . . 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 996 . . . . . 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 20187 . . . . 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 20136 . . . . . . . 8  |-  ( ( L  e.  ( UFil `  Y )  /\  g  e.  ( Fil `  Y
)  /\  L  C_  g
)  ->  L  =  g )
1612, 13, 14, 15syl3anc 1223 . . . . . . 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 5861 . . . . . 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 2504 . . . . 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 2952 . . . 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 2871 . 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 20137 . 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 968    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   -->wf 5575   ` cfv 5579  (class class class)co 6275   fBascfbas 18170   Filcfil 20074   UFilcufil 20128    FilMap cfm 20162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-fin 7510  df-fi 7860  df-fbas 18180  df-fg 18181  df-fil 20075  df-ufil 20130  df-fm 20167
This theorem is referenced by:  ufldom  20191  uffcfflf  20268
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