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Theorem fmfnfmlem1 17988
Description: Lemma for fmfnfm 17992. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Hypotheses
Ref Expression
fmfnfm.b  |-  ( ph  ->  B  e.  ( fBas `  Y ) )
fmfnfm.l  |-  ( ph  ->  L  e.  ( Fil `  X ) )
fmfnfm.f  |-  ( ph  ->  F : Y --> X )
fmfnfm.fm  |-  ( ph  ->  ( ( X  FilMap  F ) `  B ) 
C_  L )
Assertion
Ref Expression
fmfnfmlem1  |-  ( ph  ->  ( s  e.  ( fi `  B )  ->  ( ( F
" s )  C_  t  ->  ( t  C_  X  ->  t  e.  L
) ) ) )
Distinct variable groups:    t, s, B    F, s, t    L, s, t    ph, s, t    X, s, t    Y, s, t

Proof of Theorem fmfnfmlem1
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 fmfnfm.b . . . . 5  |-  ( ph  ->  B  e.  ( fBas `  Y ) )
2 fbssfi 17871 . . . . 5  |-  ( ( B  e.  ( fBas `  Y )  /\  s  e.  ( fi `  B
) )  ->  E. w  e.  B  w  C_  s
)
31, 2sylan 459 . . . 4  |-  ( (
ph  /\  s  e.  ( fi `  B ) )  ->  E. w  e.  B  w  C_  s
)
4 imass2 5242 . . . . . . 7  |-  ( w 
C_  s  ->  ( F " w )  C_  ( F " s ) )
5 sstr2 3357 . . . . . . 7  |-  ( ( F " w ) 
C_  ( F "
s )  ->  (
( F " s
)  C_  t  ->  ( F " w ) 
C_  t ) )
64, 5syl 16 . . . . . 6  |-  ( w 
C_  s  ->  (
( F " s
)  C_  t  ->  ( F " w ) 
C_  t ) )
76com12 30 . . . . 5  |-  ( ( F " s ) 
C_  t  ->  (
w  C_  s  ->  ( F " w ) 
C_  t ) )
87reximdv 2819 . . . 4  |-  ( ( F " s ) 
C_  t  ->  ( E. w  e.  B  w  C_  s  ->  E. w  e.  B  ( F " w )  C_  t
) )
93, 8syl5com 29 . . 3  |-  ( (
ph  /\  s  e.  ( fi `  B ) )  ->  ( ( F " s )  C_  t  ->  E. w  e.  B  ( F " w ) 
C_  t ) )
10 fmfnfm.l . . . . . . . 8  |-  ( ph  ->  L  e.  ( Fil `  X ) )
11 filtop 17889 . . . . . . . 8  |-  ( L  e.  ( Fil `  X
)  ->  X  e.  L )
1210, 11syl 16 . . . . . . 7  |-  ( ph  ->  X  e.  L )
13 fmfnfm.f . . . . . . 7  |-  ( ph  ->  F : Y --> X )
14 elfm 17981 . . . . . . 7  |-  ( ( X  e.  L  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( t  e.  ( ( X  FilMap  F ) `
 B )  <->  ( t  C_  X  /\  E. w  e.  B  ( F " w )  C_  t
) ) )
1512, 1, 13, 14syl3anc 1185 . . . . . 6  |-  ( ph  ->  ( t  e.  ( ( X  FilMap  F ) `
 B )  <->  ( t  C_  X  /\  E. w  e.  B  ( F " w )  C_  t
) ) )
16 fmfnfm.fm . . . . . . 7  |-  ( ph  ->  ( ( X  FilMap  F ) `  B ) 
C_  L )
1716sseld 3349 . . . . . 6  |-  ( ph  ->  ( t  e.  ( ( X  FilMap  F ) `
 B )  -> 
t  e.  L ) )
1815, 17sylbird 228 . . . . 5  |-  ( ph  ->  ( ( t  C_  X  /\  E. w  e.  B  ( F "
w )  C_  t
)  ->  t  e.  L ) )
1918exp3acom23 1382 . . . 4  |-  ( ph  ->  ( E. w  e.  B  ( F "
w )  C_  t  ->  ( t  C_  X  ->  t  e.  L ) ) )
2019adantr 453 . . 3  |-  ( (
ph  /\  s  e.  ( fi `  B ) )  ->  ( E. w  e.  B  ( F " w )  C_  t  ->  ( t  C_  X  ->  t  e.  L
) ) )
219, 20syld 43 . 2  |-  ( (
ph  /\  s  e.  ( fi `  B ) )  ->  ( ( F " s )  C_  t  ->  ( t  C_  X  ->  t  e.  L
) ) )
2221ex 425 1  |-  ( ph  ->  ( s  e.  ( fi `  B )  ->  ( ( F
" s )  C_  t  ->  ( t  C_  X  ->  t  e.  L
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726   E.wrex 2708    C_ wss 3322   "cima 4883   -->wf 5452   ` cfv 5456  (class class class)co 6083   ficfi 7417   fBascfbas 16691   Filcfil 17879    FilMap cfm 17967
This theorem is referenced by:  fmfnfmlem4  17991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-fin 7115  df-fi 7418  df-fbas 16701  df-fg 16702  df-fil 17880  df-fm 17972
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