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Theorem imaelfm 19527
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 5183 . . . . 5  |-  ( F
" S )  C_  ran  F
2 frn 5568 . . . . 5  |-  ( F : Y --> X  ->  ran  F  C_  X )
31, 2syl5ss 3370 . . . 4  |-  ( F : Y --> X  -> 
( F " S
)  C_  X )
433ad2ant3 1011 . . 3  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( F " S
)  C_  X )
5 ssid 3378 . . . 4  |-  ( F
" S )  C_  ( F " S )
6 imaeq2 5168 . . . . . 6  |-  ( x  =  S  ->  ( F " x )  =  ( F " S
) )
76sseq1d 3386 . . . . 5  |-  ( x  =  S  ->  (
( F " x
)  C_  ( F " S )  <->  ( F " S )  C_  ( F " S ) ) )
87rspcev 3076 . . . 4  |-  ( ( S  e.  L  /\  ( F " S ) 
C_  ( F " S ) )  ->  E. x  e.  L  ( F " x ) 
C_  ( F " S ) )
95, 8mpan2 671 . . 3  |-  ( S  e.  L  ->  E. x  e.  L  ( F " x )  C_  ( F " S ) )
104, 9anim12i 566 . 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 19524 . . 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 465 . 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 232 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2719    C_ wss 3331   ran crn 4844   "cima 4846   -->wf 5417   ` cfv 5421  (class class class)co 6094   fBascfbas 17807   filGencfg 17808    FilMap cfm 19509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-fbas 17817  df-fg 17818  df-fm 19514
This theorem is referenced by:  rnelfm  19529  fmfnfmlem2  19531  fmfnfmlem4  19533  fmfnfm  19534  fmco  19537  isfcf  19610  cnextcn  19642
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