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Theorem fmval 17638
Description: Introduce a function that takes a function from a filtered domain to a set and produces a filter which consists of supersets of images of filter elements. The functions which are dealt with by this function are similar to nets in topology. For example, suppose we have a sequence filtered by the filter generated by its tails under the usual natural number ordering. Then the elements of this filter are precisely the supersets of tails of this sequence. Under this definition, it is not too difficult to see that the limit of a function in the filter sense captures the notion of convergence of a sequence. As a result, the notion of a filter generalizes many ideas associated with sequences, and this function is one way to make that relationship precise in Metamath. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
fmval  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( X filGen ran  ( y  e.  B  |->  ( F " y
) ) ) )
Distinct variable groups:    y, B    y, F    y, X    y, Y    y, A

Proof of Theorem fmval
Dummy variables  f 
b  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fm 17633 . . . . 5  |-  FilMap  =  ( x  e.  _V , 
f  e.  _V  |->  ( b  e.  ( fBas `  dom  f )  |->  ( x filGen ran  ( y  e.  b  |->  ( f
" y ) ) ) ) )
21a1i 10 . . . 4  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  FilMap  =  ( x  e. 
_V ,  f  e. 
_V  |->  ( b  e.  ( fBas `  dom  f )  |->  ( x
filGen ran  ( y  e.  b  |->  ( f "
y ) ) ) ) ) )
3 dmeq 4879 . . . . . . . 8  |-  ( f  =  F  ->  dom  f  =  dom  F )
43fveq2d 5529 . . . . . . 7  |-  ( f  =  F  ->  ( fBas `  dom  f )  =  ( fBas `  dom  F ) )
54adantl 452 . . . . . 6  |-  ( ( x  =  X  /\  f  =  F )  ->  ( fBas `  dom  f )  =  (
fBas `  dom  F ) )
6 id 19 . . . . . . 7  |-  ( x  =  X  ->  x  =  X )
7 imaeq1 5007 . . . . . . . . 9  |-  ( f  =  F  ->  (
f " y )  =  ( F "
y ) )
87mpteq2dv 4107 . . . . . . . 8  |-  ( f  =  F  ->  (
y  e.  b  |->  ( f " y ) )  =  ( y  e.  b  |->  ( F
" y ) ) )
98rneqd 4906 . . . . . . 7  |-  ( f  =  F  ->  ran  ( y  e.  b 
|->  ( f " y
) )  =  ran  ( y  e.  b 
|->  ( F " y
) ) )
106, 9oveqan12d 5877 . . . . . 6  |-  ( ( x  =  X  /\  f  =  F )  ->  ( x filGen ran  (
y  e.  b  |->  ( f " y ) ) )  =  ( X filGen ran  ( y  e.  b  |->  ( F
" y ) ) ) )
115, 10mpteq12dv 4098 . . . . 5  |-  ( ( x  =  X  /\  f  =  F )  ->  ( b  e.  (
fBas `  dom  f ) 
|->  ( x filGen ran  (
y  e.  b  |->  ( f " y ) ) ) )  =  ( b  e.  (
fBas `  dom  F ) 
|->  ( X filGen ran  (
y  e.  b  |->  ( F " y ) ) ) ) )
12 fdm 5393 . . . . . . . 8  |-  ( F : Y --> X  ->  dom  F  =  Y )
1312fveq2d 5529 . . . . . . 7  |-  ( F : Y --> X  -> 
( fBas `  dom  F )  =  ( fBas `  Y
) )
14 mpteq1 4100 . . . . . . 7  |-  ( (
fBas `  dom  F )  =  ( fBas `  Y
)  ->  ( b  e.  ( fBas `  dom  F )  |->  ( X filGen ran  ( y  e.  b 
|->  ( F " y
) ) ) )  =  ( b  e.  ( fBas `  Y
)  |->  ( X filGen ran  ( y  e.  b 
|->  ( F " y
) ) ) ) )
1513, 14syl 15 . . . . . 6  |-  ( F : Y --> X  -> 
( b  e.  (
fBas `  dom  F ) 
|->  ( X filGen ran  (
y  e.  b  |->  ( F " y ) ) ) )  =  ( b  e.  (
fBas `  Y )  |->  ( X filGen ran  (
y  e.  b  |->  ( F " y ) ) ) ) )
16153ad2ant3 978 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( b  e.  (
fBas `  dom  F ) 
|->  ( X filGen ran  (
y  e.  b  |->  ( F " y ) ) ) )  =  ( b  e.  (
fBas `  Y )  |->  ( X filGen ran  (
y  e.  b  |->  ( F " y ) ) ) ) )
1711, 16sylan9eqr 2337 . . . 4  |-  ( ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  ( x  =  X  /\  f  =  F ) )  -> 
( b  e.  (
fBas `  dom  f ) 
|->  ( x filGen ran  (
y  e.  b  |->  ( f " y ) ) ) )  =  ( b  e.  (
fBas `  Y )  |->  ( X filGen ran  (
y  e.  b  |->  ( F " y ) ) ) ) )
18 elex 2796 . . . . 5  |-  ( X  e.  A  ->  X  e.  _V )
19183ad2ant1 976 . . . 4  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  X  e.  _V )
20 simp3 957 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  F : Y --> X )
21 elfvdm 5554 . . . . . 6  |-  ( B  e.  ( fBas `  Y
)  ->  Y  e.  dom  fBas )
22213ad2ant2 977 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  Y  e.  dom  fBas )
23 simp1 955 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  X  e.  A )
24 fex2 5401 . . . . 5  |-  ( ( F : Y --> X  /\  Y  e.  dom  fBas  /\  X  e.  A )  ->  F  e.  _V )
2520, 22, 23, 24syl3anc 1182 . . . 4  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  F  e.  _V )
26 fvex 5539 . . . . . 6  |-  ( fBas `  Y )  e.  _V
2726mptex 5746 . . . . 5  |-  ( b  e.  ( fBas `  Y
)  |->  ( X filGen ran  ( y  e.  b 
|->  ( F " y
) ) ) )  e.  _V
2827a1i 10 . . . 4  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( b  e.  (
fBas `  Y )  |->  ( X filGen ran  (
y  e.  b  |->  ( F " y ) ) ) )  e. 
_V )
292, 17, 19, 25, 28ovmpt2d 5975 . . 3  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( X  FilMap  F )  =  ( b  e.  ( fBas `  Y
)  |->  ( X filGen ran  ( y  e.  b 
|->  ( F " y
) ) ) ) )
3029fveq1d 5527 . 2  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( ( b  e.  ( fBas `  Y
)  |->  ( X filGen ran  ( y  e.  b 
|->  ( F " y
) ) ) ) `
 B ) )
31 mpteq1 4100 . . . . . 6  |-  ( b  =  B  ->  (
y  e.  b  |->  ( F " y ) )  =  ( y  e.  B  |->  ( F
" y ) ) )
3231rneqd 4906 . . . . 5  |-  ( b  =  B  ->  ran  ( y  e.  b 
|->  ( F " y
) )  =  ran  ( y  e.  B  |->  ( F " y
) ) )
3332oveq2d 5874 . . . 4  |-  ( b  =  B  ->  ( X filGen ran  ( y  e.  b  |->  ( F
" y ) ) )  =  ( X
filGen ran  ( y  e.  B  |->  ( F "
y ) ) ) )
34 eqid 2283 . . . 4  |-  ( b  e.  ( fBas `  Y
)  |->  ( X filGen ran  ( y  e.  b 
|->  ( F " y
) ) ) )  =  ( b  e.  ( fBas `  Y
)  |->  ( X filGen ran  ( y  e.  b 
|->  ( F " y
) ) ) )
35 ovex 5883 . . . 4  |-  ( X
filGen ran  ( y  e.  B  |->  ( F "
y ) ) )  e.  _V
3633, 34, 35fvmpt 5602 . . 3  |-  ( B  e.  ( fBas `  Y
)  ->  ( (
b  e.  ( fBas `  Y )  |->  ( X
filGen ran  ( y  e.  b  |->  ( F "
y ) ) ) ) `  B )  =  ( X filGen ran  ( y  e.  B  |->  ( F " y
) ) ) )
37363ad2ant2 977 . 2  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( b  e.  ( fBas `  Y
)  |->  ( X filGen ran  ( y  e.  b 
|->  ( F " y
) ) ) ) `
 B )  =  ( X filGen ran  (
y  e.  B  |->  ( F " y ) ) ) )
3830, 37eqtrd 2315 1  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( X filGen ran  ( y  e.  B  |->  ( F " y
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   dom cdm 4689   ran crn 4690   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   fBascfbas 17518   filGencfg 17519    FilMap cfm 17628
This theorem is referenced by:  fmfil  17639  fmss  17641  elfm  17642  fmcfil  18698
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-fm 17633
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