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Theorem fmfil 20180
Description: A mapping filter is a filter. (Contributed by Jeff Hankins, 18-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
fmfil  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  e.  ( Fil `  X
) )

Proof of Theorem fmfil
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fmval 20179 . 2  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( X filGen ran  ( y  e.  B  |->  ( F " y
) ) ) )
2 eqid 2467 . . . . 5  |-  ran  (
y  e.  B  |->  ( F " y ) )  =  ran  (
y  e.  B  |->  ( F " y ) )
32fbasrn 20120 . . . 4  |-  ( ( B  e.  ( fBas `  Y )  /\  F : Y --> X  /\  X  e.  A )  ->  ran  ( y  e.  B  |->  ( F " y
) )  e.  (
fBas `  X )
)
433comr 1204 . . 3  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  ran  ( y  e.  B  |->  ( F " y
) )  e.  (
fBas `  X )
)
5 fgcl 20114 . . 3  |-  ( ran  ( y  e.  B  |->  ( F " y
) )  e.  (
fBas `  X )  ->  ( X filGen ran  (
y  e.  B  |->  ( F " y ) ) )  e.  ( Fil `  X ) )
64, 5syl 16 . 2  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( X filGen ran  (
y  e.  B  |->  ( F " y ) ) )  e.  ( Fil `  X ) )
71, 6eqeltrd 2555 1  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  e.  ( Fil `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    e. wcel 1767    |-> cmpt 4505   ran crn 5000   "cima 5002   -->wf 5582   ` cfv 5586  (class class class)co 6282   fBascfbas 18177   filGencfg 18178   Filcfil 20081    FilMap cfm 20169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-fbas 18187  df-fg 18188  df-fil 20082  df-fm 20174
This theorem is referenced by:  fmf  20181  fmufil  20195  fmco  20197  ufldom  20198  flfnei  20227  isflf  20229  flfcnp  20240  isfcf  20270  cnpfcfi  20276  cnpfcf  20277  cnextucn  20541
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