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

Theorem fmval 17980
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 17975 . . . . 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 5073 . . . . . . . 8  |-  ( f  =  F  ->  dom  f  =  dom  F )
43fveq2d 5735 . . . . . . 7  |-  ( f  =  F  ->  ( fBas `  dom  f )  =  ( fBas `  dom  F ) )
54adantl 454 . . . . . 6  |-  ( ( x  =  X  /\  f  =  F )  ->  ( fBas `  dom  f )  =  (
fBas `  dom  F ) )
6 id 21 . . . . . . 7  |-  ( x  =  X  ->  x  =  X )
7 imaeq1 5201 . . . . . . . . 9  |-  ( f  =  F  ->  (
f " y )  =  ( F "
y ) )
87mpteq2dv 4299 . . . . . . . 8  |-  ( f  =  F  ->  (
y  e.  b  |->  ( f " y ) )  =  ( y  e.  b  |->  ( F
" y ) ) )
98rneqd 5100 . . . . . . 7  |-  ( f  =  F  ->  ran  ( y  e.  b 
|->  ( f " y
) )  =  ran  ( y  e.  b 
|->  ( F " y
) ) )
106, 9oveqan12d 6103 . . . . . 6  |-  ( ( x  =  X  /\  f  =  F )  ->  ( x filGen ran  (
y  e.  b  |->  ( f " y ) ) )  =  ( X filGen ran  ( y  e.  b  |->  ( F
" y ) ) ) )
115, 10mpteq12dv 4290 . . . . 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 5598 . . . . . . . 8  |-  ( F : Y --> X  ->  dom  F  =  Y )
1312fveq2d 5735 . . . . . . 7  |-  ( F : Y --> X  -> 
( fBas `  dom  F )  =  ( fBas `  Y
) )
1413mpteq1d 4293 . . . . . 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 981 . . . . 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 2492 . . . 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 2966 . . . . 5  |-  ( X  e.  A  ->  X  e.  _V )
18173ad2ant1 979 . . . 4  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  X  e.  _V )
19 simp3 960 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  F : Y --> X )
20 elfvdm 5760 . . . . . 6  |-  ( B  e.  ( fBas `  Y
)  ->  Y  e.  dom  fBas )
21203ad2ant2 980 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  Y  e.  dom  fBas )
22 simp1 958 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  X  e.  A )
23 fex2 5606 . . . . 5  |-  ( ( F : Y --> X  /\  Y  e.  dom  fBas  /\  X  e.  A )  ->  F  e.  _V )
2419, 21, 22, 23syl3anc 1185 . . . 4  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  F  e.  _V )
25 fvex 5745 . . . . . 6  |-  ( fBas `  Y )  e.  _V
2625mptex 5969 . . . . 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 6204 . . 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 5733 . 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 4292 . . . . . 6  |-  ( b  =  B  ->  (
y  e.  b  |->  ( F " y ) )  =  ( y  e.  B  |->  ( F
" y ) ) )
3130rneqd 5100 . . . . 5  |-  ( b  =  B  ->  ran  ( y  e.  b 
|->  ( F " y
) )  =  ran  ( y  e.  B  |->  ( F " y
) ) )
3231oveq2d 6100 . . . 4  |-  ( b  =  B  ->  ( X filGen ran  ( y  e.  b  |->  ( F
" y ) ) )  =  ( X
filGen ran  ( y  e.  B  |->  ( F "
y ) ) ) )
33 eqid 2438 . . . 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 6109 . . . 4  |-  ( X
filGen ran  ( y  e.  B  |->  ( F "
y ) ) )  e.  _V
3532, 33, 34fvmpt 5809 . . 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 980 . 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 2470 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    e. cmpt 4269   dom cdm 4881   ran crn 4882   "cima 4884   -->wf 5453   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   fBascfbas 16694   filGencfg 16695    FilMap cfm 17970
This theorem is referenced by:  fmfil  17981  fmss  17983  elfm  17984  ucnextcn  18339  fmcfil  19230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-fm 17975
  Copyright terms: Public domain W3C validator