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Theorem imaelfm 20578
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 5358 . . . . 5  |-  ( F
" S )  C_  ran  F
2 frn 5743 . . . . 5  |-  ( F : Y --> X  ->  ran  F  C_  X )
31, 2syl5ss 3510 . . . 4  |-  ( F : Y --> X  -> 
( F " S
)  C_  X )
433ad2ant3 1019 . . 3  |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( F " S
)  C_  X )
5 ssid 3518 . . . 4  |-  ( F
" S )  C_  ( F " S )
6 imaeq2 5343 . . . . . 6  |-  ( x  =  S  ->  ( F " x )  =  ( F " S
) )
76sseq1d 3526 . . . . 5  |-  ( x  =  S  ->  (
( F " x
)  C_  ( F " S )  <->  ( F " S )  C_  ( F " S ) ) )
87rspcev 3210 . . . 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 20575 . . 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 973    = wceq 1395    e. wcel 1819   E.wrex 2808    C_ wss 3471   ran crn 5009   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296   fBascfbas 18533   filGencfg 18534    FilMap cfm 20560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-fbas 18543  df-fg 18544  df-fm 20565
This theorem is referenced by:  rnelfm  20580  fmfnfmlem2  20582  fmfnfmlem4  20584  fmfnfm  20585  fmco  20588  isfcf  20661  cnextcn  20693
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