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Theorem fmval 19634
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 19629 . . . . 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 5140 . . . . . . . 8  |-  ( f  =  F  ->  dom  f  =  dom  F )
43fveq2d 5795 . . . . . . 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 5264 . . . . . . . . 9  |-  ( f  =  F  ->  (
f " y )  =  ( F "
y ) )
87mpteq2dv 4479 . . . . . . . 8  |-  ( f  =  F  ->  (
y  e.  b  |->  ( f " y ) )  =  ( y  e.  b  |->  ( F
" y ) ) )
98rneqd 5167 . . . . . . 7  |-  ( f  =  F  ->  ran  ( y  e.  b 
|->  ( f " y
) )  =  ran  ( y  e.  b 
|->  ( F " y
) ) )
106, 9oveqan12d 6211 . . . . . 6  |-  ( ( x  =  X  /\  f  =  F )  ->  ( x filGen ran  (
y  e.  b  |->  ( f " y ) ) )  =  ( X filGen ran  ( y  e.  b  |->  ( F
" y ) ) ) )
115, 10mpteq12dv 4470 . . . . 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 5663 . . . . . . . 8  |-  ( F : Y --> X  ->  dom  F  =  Y )
1312fveq2d 5795 . . . . . . 7  |-  ( F : Y --> X  -> 
( fBas `  dom  F )  =  ( fBas `  Y
) )
1413mpteq1d 4473 . . . . . 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 1011 . . . . 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 2514 . . . 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 3079 . . . . 5  |-  ( X  e.  A  ->  X  e.  _V )
18173ad2ant1 1009 . . . 4  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  X  e.  _V )
19 simp3 990 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  F : Y --> X )
20 elfvdm 5817 . . . . . 6  |-  ( B  e.  ( fBas `  Y
)  ->  Y  e.  dom  fBas )
21203ad2ant2 1010 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  Y  e.  dom  fBas )
22 simp1 988 . . . . 5  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  X  e.  A )
23 fex2 6634 . . . . 5  |-  ( ( F : Y --> X  /\  Y  e.  dom  fBas  /\  X  e.  A )  ->  F  e.  _V )
2419, 21, 22, 23syl3anc 1219 . . . 4  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  F  e.  _V )
25 fvex 5801 . . . . . 6  |-  ( fBas `  Y )  e.  _V
2625mptex 6049 . . . . 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 6320 . . 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 5793 . 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 4472 . . . . . 6  |-  ( b  =  B  ->  (
y  e.  b  |->  ( F " y ) )  =  ( y  e.  B  |->  ( F
" y ) ) )
3130rneqd 5167 . . . . 5  |-  ( b  =  B  ->  ran  ( y  e.  b 
|->  ( F " y
) )  =  ran  ( y  e.  B  |->  ( F " y
) ) )
3231oveq2d 6208 . . . 4  |-  ( b  =  B  ->  ( X filGen ran  ( y  e.  b  |->  ( F
" y ) ) )  =  ( X
filGen ran  ( y  e.  B  |->  ( F "
y ) ) ) )
33 eqid 2451 . . . 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 6217 . . . 4  |-  ( X
filGen ran  ( y  e.  B  |->  ( F "
y ) ) )  e.  _V
3532, 33, 34fvmpt 5875 . . 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 1010 . 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 2492 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 965    = wceq 1370    e. wcel 1758   _Vcvv 3070    |-> cmpt 4450   dom cdm 4940   ran crn 4941   "cima 4943   -->wf 5514   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   fBascfbas 17915   filGencfg 17916    FilMap cfm 19624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  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 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-fm 19629
This theorem is referenced by:  fmfil  19635  fmss  19637  elfm  19638  ucnextcn  19997  fmcfil  20901
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