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Theorem fmval 20172
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 positive integer 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 20167 . . . . 5  |-  FilMap  =  ( x  e.  _V , 
f  e.  _V  |->  ( b  e.  ( fBas `  dom  f )  |->  ( x filGen ran  ( y  e.  b  |->  ( f
" y ) ) ) ) )
21a1i 11 . . . 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 5194 . . . . . . . 8  |-  ( f  =  F  ->  dom  f  =  dom  F )
43fveq2d 5861 . . . . . . 7  |-  ( f  =  F  ->  ( fBas `  dom  f )  =  ( fBas `  dom  F ) )
54adantl 466 . . . . . 6  |-  ( ( x  =  X  /\  f  =  F )  ->  ( fBas `  dom  f )  =  (
fBas `  dom  F ) )
6 id 22 . . . . . . 7  |-  ( x  =  X  ->  x  =  X )
7 imaeq1 5323 . . . . . . . . 9  |-  ( f  =  F  ->  (
f " y )  =  ( F "
y ) )
87mpteq2dv 4527 . . . . . . . 8  |-  ( f  =  F  ->  (
y  e.  b  |->  ( f " y ) )  =  ( y  e.  b  |->  ( F
" y ) ) )
98rneqd 5221 . . . . . . 7  |-  ( f  =  F  ->  ran  ( y  e.  b 
|->  ( f " y
) )  =  ran  ( y  e.  b 
|->  ( F " y
) ) )
106, 9oveqan12d 6294 . . . . . 6  |-  ( ( x  =  X  /\  f  =  F )  ->  ( x filGen ran  (
y  e.  b  |->  ( f " y ) ) )  =  ( X filGen ran  ( y  e.  b  |->  ( F
" y ) ) ) )
115, 10mpteq12dv 4518 . . . . 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 5726 . . . . . . . 8  |-  ( F : Y --> X  ->  dom  F  =  Y )
1312fveq2d 5861 . . . . . . 7  |-  ( F : Y --> X  -> 
( fBas `  dom  F )  =  ( fBas `  Y
) )
1413mpteq1d 4521 . . . . . 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 ) ) ) ) )
15143ad2ant3 1014 . . . . 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 ) ) ) ) )
1611, 15sylan9eqr 2523 . . . 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 ) ) ) ) )
17 elex 3115 . . . . 5  |-  ( X  e.  A  ->  X  e.  _V )
18173ad2ant1 1012 . . . 4  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  X  e.  _V )
19 simp3 993 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  F : Y --> X )
20 elfvdm 5883 . . . . . 6  |-  ( B  e.  ( fBas `  Y
)  ->  Y  e.  dom  fBas )
21203ad2ant2 1013 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  Y  e.  dom  fBas )
22 simp1 991 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  X  e.  A )
23 fex2 6729 . . . . 5  |-  ( ( F : Y --> X  /\  Y  e.  dom  fBas  /\  X  e.  A )  ->  F  e.  _V )
2419, 21, 22, 23syl3anc 1223 . . . 4  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  F  e.  _V )
25 fvex 5867 . . . . . 6  |-  ( fBas `  Y )  e.  _V
2625mptex 6122 . . . . 5  |-  ( b  e.  ( fBas `  Y
)  |->  ( X filGen ran  ( y  e.  b 
|->  ( F " y
) ) ) )  e.  _V
2726a1i 11 . . . 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 )
282, 16, 18, 24, 27ovmpt2d 6405 . . 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
) ) ) ) )
2928fveq1d 5859 . 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 ) )
30 mpteq1 4520 . . . . . 6  |-  ( b  =  B  ->  (
y  e.  b  |->  ( F " y ) )  =  ( y  e.  B  |->  ( F
" y ) ) )
3130rneqd 5221 . . . . 5  |-  ( b  =  B  ->  ran  ( y  e.  b 
|->  ( F " y
) )  =  ran  ( y  e.  B  |->  ( F " y
) ) )
3231oveq2d 6291 . . . 4  |-  ( b  =  B  ->  ( X filGen ran  ( y  e.  b  |->  ( F
" y ) ) )  =  ( X
filGen ran  ( y  e.  B  |->  ( F "
y ) ) ) )
33 eqid 2460 . . . 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
) ) ) )
34 ovex 6300 . . . 4  |-  ( X
filGen ran  ( y  e.  B  |->  ( F "
y ) ) )  e.  _V
3532, 33, 34fvmpt 5941 . . 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
) ) ) )
36353ad2ant2 1013 . 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 ) ) ) )
3729, 36eqtrd 2501 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3106    |-> cmpt 4498   dom cdm 4992   ran crn 4993   "cima 4995   -->wf 5575   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   fBascfbas 18170   filGencfg 18171    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-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-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-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  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-fm 20167
This theorem is referenced by:  fmfil  20173  fmss  20175  elfm  20176  ucnextcn  20535  fmcfil  21439
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