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Theorem imaelfm 17646
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 5025 . . . . 5  |-  ( F
" S )  C_  ran  F
2 frn 5395 . . . . 5  |-  ( F : Y --> X  ->  ran  F  C_  X )
31, 2syl5ss 3190 . . . 4  |-  ( F : Y --> X  -> 
( F " S
)  C_  X )
433ad2ant3 978 . . 3  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( F " S
)  C_  X )
5 ssid 3197 . . . 4  |-  ( F
" S )  C_  ( F " S )
6 imaeq2 5008 . . . . . 6  |-  ( x  =  S  ->  ( F " x )  =  ( F " S
) )
76sseq1d 3205 . . . . 5  |-  ( x  =  S  ->  (
( F " x
)  C_  ( F " S )  <->  ( F " S )  C_  ( F " S ) ) )
87rspcev 2884 . . . 4  |-  ( ( S  e.  L  /\  ( F " S ) 
C_  ( F " S ) )  ->  E. x  e.  L  ( F " x ) 
C_  ( F " S ) )
95, 8mpan2 652 . . 3  |-  ( S  e.  L  ->  E. x  e.  L  ( F " x )  C_  ( F " S ) )
104, 9anim12i 549 . 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 17643 . . 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 451 . 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 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ran crn 4690   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   fBascfbas 17518   filGencfg 17519    FilMap cfm 17628
This theorem is referenced by:  rnelfm  17648  fmfnfmlem2  17650  fmfnfmlem4  17652  fmfnfm  17653  fmco  17656  isfcf  17729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-fbas 17520  df-fg 17521  df-fm 17633
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