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Theorem fmfnfmlem1 17649
Description: Lemma for fmfnfm 17653. (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 17532 . . . . 5  |-  ( ( B  e.  ( fBas `  Y )  /\  s  e.  ( fi `  B
) )  ->  E. w  e.  B  w  C_  s
)
31, 2sylan 457 . . . 4  |-  ( (
ph  /\  s  e.  ( fi `  B ) )  ->  E. w  e.  B  w  C_  s
)
4 imass2 5049 . . . . . . 7  |-  ( w 
C_  s  ->  ( F " w )  C_  ( F " s ) )
5 sstr2 3186 . . . . . . 7  |-  ( ( F " w ) 
C_  ( F "
s )  ->  (
( F " s
)  C_  t  ->  ( F " w ) 
C_  t ) )
64, 5syl 15 . . . . . 6  |-  ( w 
C_  s  ->  (
( F " s
)  C_  t  ->  ( F " w ) 
C_  t ) )
76com12 27 . . . . 5  |-  ( ( F " s ) 
C_  t  ->  (
w  C_  s  ->  ( F " w ) 
C_  t ) )
87reximdv 2654 . . . 4  |-  ( ( F " s ) 
C_  t  ->  ( E. w  e.  B  w  C_  s  ->  E. w  e.  B  ( F " w )  C_  t
) )
93, 8syl5com 26 . . 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 17550 . . . . . . . 8  |-  ( L  e.  ( Fil `  X
)  ->  X  e.  L )
1210, 11syl 15 . . . . . . 7  |-  ( ph  ->  X  e.  L )
13 fmfnfm.f . . . . . . 7  |-  ( ph  ->  F : Y --> X )
14 elfm 17642 . . . . . . 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 1182 . . . . . 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 3179 . . . . . 6  |-  ( ph  ->  ( t  e.  ( ( X  FilMap  F ) `
 B )  -> 
t  e.  L ) )
1815, 17sylbird 226 . . . . 5  |-  ( ph  ->  ( ( t  C_  X  /\  E. w  e.  B  ( F "
w )  C_  t
)  ->  t  e.  L ) )
1918exp3acom23 1362 . . . 4  |-  ( ph  ->  ( E. w  e.  B  ( F "
w )  C_  t  ->  ( t  C_  X  ->  t  e.  L ) ) )
2019adantr 451 . . 3  |-  ( (
ph  /\  s  e.  ( fi `  B ) )  ->  ( E. w  e.  B  ( F " w )  C_  t  ->  ( t  C_  X  ->  t  e.  L
) ) )
219, 20syld 40 . 2  |-  ( (
ph  /\  s  e.  ( fi `  B ) )  ->  ( ( F " s )  C_  t  ->  ( t  C_  X  ->  t  e.  L
) ) )
2221ex 423 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 176    /\ wa 358    e. wcel 1684   E.wrex 2544    C_ wss 3152   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   ficfi 7164   fBascfbas 17518   Filcfil 17540    FilMap cfm 17628
This theorem is referenced by:  fmfnfmlem4  17652
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-3or 935  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633
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