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Theorem fmfnfmlem1 17665
Description: Lemma for fmfnfm 17669. (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 17548 . . . . 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 5065 . . . . . . 7  |-  ( w 
C_  s  ->  ( F " w )  C_  ( F " s ) )
5 sstr2 3199 . . . . . . 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 2667 . . . 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 17566 . . . . . . . 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 17658 . . . . . . 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 3192 . . . . . 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 1696   E.wrex 2557    C_ wss 3165   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   ficfi 7180   fBascfbas 17534   Filcfil 17556    FilMap cfm 17644
This theorem is referenced by:  fmfnfmlem4  17668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-fin 6883  df-fi 7181  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649
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