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Theorem imaelfm 17983
Description: An image of a filter element is in the image filter. (Contributed by Jeff Hankins, 5-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
imaelfm.l  |-  L  =  ( Y filGen B )
Assertion
Ref Expression
imaelfm  |-  ( ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  S  e.  L
)  ->  ( F " S )  e.  ( ( X  FilMap  F ) `
 B ) )

Proof of Theorem imaelfm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 imassrn 5216 . . . . 5  |-  ( F
" S )  C_  ran  F
2 frn 5597 . . . . 5  |-  ( F : Y --> X  ->  ran  F  C_  X )
31, 2syl5ss 3359 . . . 4  |-  ( F : Y --> X  -> 
( F " S
)  C_  X )
433ad2ant3 980 . . 3  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( F " S
)  C_  X )
5 ssid 3367 . . . 4  |-  ( F
" S )  C_  ( F " S )
6 imaeq2 5199 . . . . . 6  |-  ( x  =  S  ->  ( F " x )  =  ( F " S
) )
76sseq1d 3375 . . . . 5  |-  ( x  =  S  ->  (
( F " x
)  C_  ( F " S )  <->  ( F " S )  C_  ( F " S ) ) )
87rspcev 3052 . . . 4  |-  ( ( S  e.  L  /\  ( F " S ) 
C_  ( F " S ) )  ->  E. x  e.  L  ( F " x ) 
C_  ( F " S ) )
95, 8mpan2 653 . . 3  |-  ( S  e.  L  ->  E. x  e.  L  ( F " x )  C_  ( F " S ) )
104, 9anim12i 550 . 2  |-  ( ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  S  e.  L
)  ->  ( ( F " S )  C_  X  /\  E. x  e.  L  ( F "
x )  C_  ( F " S ) ) )
11 imaelfm.l . . . 4  |-  L  =  ( Y filGen B )
1211elfm2 17980 . . 3  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( F " S )  e.  ( ( X  FilMap  F ) `
 B )  <->  ( ( F " S )  C_  X  /\  E. x  e.  L  ( F "
x )  C_  ( F " S ) ) ) )
1312adantr 452 . 2  |-  ( ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  S  e.  L
)  ->  ( ( F " S )  e.  ( ( X  FilMap  F ) `  B )  <-> 
( ( F " S )  C_  X  /\  E. x  e.  L  ( F " x ) 
C_  ( F " S ) ) ) )
1410, 13mpbird 224 1  |-  ( ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  S  e.  L
)  ->  ( F " S )  e.  ( ( X  FilMap  F ) `
 B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706    C_ wss 3320   ran crn 4879   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081   fBascfbas 16689   filGencfg 16690    FilMap cfm 17965
This theorem is referenced by:  rnelfm  17985  fmfnfmlem2  17987  fmfnfmlem4  17989  fmfnfm  17990  fmco  17993  isfcf  18066  cnextcn  18098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-fbas 16699  df-fg 16700  df-fm 17970
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