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Theorem fmval 17734
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 17729 . . . . 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 4958 . . . . . . . 8  |-  ( f  =  F  ->  dom  f  =  dom  F )
43fveq2d 5609 . . . . . . 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 5086 . . . . . . . . 9  |-  ( f  =  F  ->  (
f " y )  =  ( F "
y ) )
87mpteq2dv 4186 . . . . . . . 8  |-  ( f  =  F  ->  (
y  e.  b  |->  ( f " y ) )  =  ( y  e.  b  |->  ( F
" y ) ) )
98rneqd 4985 . . . . . . 7  |-  ( f  =  F  ->  ran  ( y  e.  b 
|->  ( f " y
) )  =  ran  ( y  e.  b 
|->  ( F " y
) ) )
106, 9oveqan12d 5961 . . . . . 6  |-  ( ( x  =  X  /\  f  =  F )  ->  ( x filGen ran  (
y  e.  b  |->  ( f " y ) ) )  =  ( X filGen ran  ( y  e.  b  |->  ( F
" y ) ) ) )
115, 10mpteq12dv 4177 . . . . 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 5473 . . . . . . . 8  |-  ( F : Y --> X  ->  dom  F  =  Y )
1312fveq2d 5609 . . . . . . 7  |-  ( F : Y --> X  -> 
( fBas `  dom  F )  =  ( fBas `  Y
) )
14 mpteq1 4179 . . . . . . 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 2412 . . . 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 2872 . . . . 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 5634 . . . . . 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 5481 . . . . 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 5619 . . . . . 6  |-  ( fBas `  Y )  e.  _V
2726mptex 5829 . . . . 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 6059 . . 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 5607 . 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 4179 . . . . . 6  |-  ( b  =  B  ->  (
y  e.  b  |->  ( F " y ) )  =  ( y  e.  B  |->  ( F
" y ) ) )
3231rneqd 4985 . . . . 5  |-  ( b  =  B  ->  ran  ( y  e.  b 
|->  ( F " y
) )  =  ran  ( y  e.  B  |->  ( F " y
) ) )
3332oveq2d 5958 . . . 4  |-  ( b  =  B  ->  ( X filGen ran  ( y  e.  b  |->  ( F
" y ) ) )  =  ( X
filGen ran  ( y  e.  B  |->  ( F "
y ) ) ) )
34 eqid 2358 . . . 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 5967 . . . 4  |-  ( X
filGen ran  ( y  e.  B  |->  ( F "
y ) ) )  e.  _V
3633, 34, 35fvmpt 5682 . . 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 2390 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 1642    e. wcel 1710   _Vcvv 2864    e. cmpt 4156   dom cdm 4768   ran crn 4769   "cima 4771   -->wf 5330   ` cfv 5334  (class class class)co 5942    e. cmpt2 5944   fBascfbas 16465   filGencfg 16466    FilMap cfm 17724
This theorem is referenced by:  fmfil  17735  fmss  17737  elfm  17738  fmcfil  18796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-fm 17729
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