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Theorem imaelfm 19366
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 5168 . . . . 5  |-  ( F
" S )  C_  ran  F
2 frn 5553 . . . . 5  |-  ( F : Y --> X  ->  ran  F  C_  X )
31, 2syl5ss 3355 . . . 4  |-  ( F : Y --> X  -> 
( F " S
)  C_  X )
433ad2ant3 1004 . . 3  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( F " S
)  C_  X )
5 ssid 3363 . . . 4  |-  ( F
" S )  C_  ( F " S )
6 imaeq2 5153 . . . . . 6  |-  ( x  =  S  ->  ( F " x )  =  ( F " S
) )
76sseq1d 3371 . . . . 5  |-  ( x  =  S  ->  (
( F " x
)  C_  ( F " S )  <->  ( F " S )  C_  ( F " S ) ) )
87rspcev 3062 . . . 4  |-  ( ( S  e.  L  /\  ( F " S ) 
C_  ( F " S ) )  ->  E. x  e.  L  ( F " x ) 
C_  ( F " S ) )
95, 8mpan2 664 . . 3  |-  ( S  e.  L  ->  E. x  e.  L  ( F " x )  C_  ( F " S ) )
104, 9anim12i 561 . 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 19363 . . 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 462 . 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 958    = wceq 1362    e. wcel 1755   E.wrex 2706    C_ wss 3316   ran crn 4828   "cima 4830   -->wf 5402   ` cfv 5406  (class class class)co 6080   fBascfbas 17648   filGencfg 17649    FilMap cfm 19348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-fbas 17658  df-fg 17659  df-fm 19353
This theorem is referenced by:  rnelfm  19368  fmfnfmlem2  19370  fmfnfmlem4  19372  fmfnfm  19373  fmco  19376  isfcf  19449  cnextcn  19481
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