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Theorem fgtr 17927
Description: If  A is a member of the filter, then truncating  F to  A and regenerating the behavior outside  A using 
filGen recovers the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
fgtr  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen ( Ft  A ) )  =  F )

Proof of Theorem fgtr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filfbas 17885 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 fbncp 17876 . . . . . . . 8  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  ( X  \  A )  e.  F )
31, 2sylan 459 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  -.  ( X  \  A )  e.  F )
4 filelss 17889 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  C_  X )
5 trfil3 17925 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  (
( Ft  A )  e.  ( Fil `  A )  <->  -.  ( X  \  A
)  e.  F ) )
64, 5syldan 458 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
( Ft  A )  e.  ( Fil `  A )  <->  -.  ( X  \  A
)  e.  F ) )
73, 6mpbird 225 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  e.  ( Fil `  A ) )
8 filfbas 17885 . . . . . 6  |-  ( ( Ft  A )  e.  ( Fil `  A )  ->  ( Ft  A )  e.  ( fBas `  A
) )
97, 8syl 16 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  e.  (
fBas `  A )
)
10 restsspw 13664 . . . . . 6  |-  ( Ft  A )  C_  ~P A
11 sspwb 4416 . . . . . . 7  |-  ( A 
C_  X  <->  ~P A  C_ 
~P X )
124, 11sylib 190 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ~P A  C_  ~P X )
1310, 12syl5ss 3361 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  C_  ~P X )
14 filtop 17892 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
1514adantr 453 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  X  e.  F )
16 fbasweak 17902 . . . . 5  |-  ( ( ( Ft  A )  e.  (
fBas `  A )  /\  ( Ft  A )  C_  ~P X  /\  X  e.  F
)  ->  ( Ft  A
)  e.  ( fBas `  X ) )
179, 13, 15, 16syl3anc 1185 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  e.  (
fBas `  X )
)
181adantr 453 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  F  e.  ( fBas `  X
) )
19 trfilss 17926 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  C_  F
)
20 fgss 17910 . . . 4  |-  ( ( ( Ft  A )  e.  (
fBas `  X )  /\  F  e.  ( fBas `  X )  /\  ( Ft  A )  C_  F
)  ->  ( X filGen ( Ft  A ) )  C_  ( X filGen F ) )
2117, 18, 19, 20syl3anc 1185 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen ( Ft  A ) )  C_  ( X filGen F ) )
22 fgfil 17912 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
2322adantr 453 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen F )  =  F )
2421, 23sseqtrd 3386 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen ( Ft  A ) )  C_  F )
25 filelss 17889 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F )  ->  x  C_  X )
2625ex 425 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  x  C_  X ) )
2726adantr 453 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  x  C_  X ) )
28 elrestr 13661 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  x  e.  F )  ->  (
x  i^i  A )  e.  ( Ft  A ) )
29283expa 1154 . . . . . . 7  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  e.  F
)  /\  x  e.  F )  ->  (
x  i^i  A )  e.  ( Ft  A ) )
30 inss1 3563 . . . . . . 7  |-  ( x  i^i  A )  C_  x
31 sseq1 3371 . . . . . . . 8  |-  ( y  =  ( x  i^i 
A )  ->  (
y  C_  x  <->  ( x  i^i  A )  C_  x
) )
3231rspcev 3054 . . . . . . 7  |-  ( ( ( x  i^i  A
)  e.  ( Ft  A )  /\  ( x  i^i  A )  C_  x )  ->  E. y  e.  ( Ft  A ) y  C_  x )
3329, 30, 32sylancl 645 . . . . . 6  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  e.  F
)  /\  x  e.  F )  ->  E. y  e.  ( Ft  A ) y  C_  x )
3433ex 425 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  E. y  e.  ( Ft  A ) y  C_  x ) )
3527, 34jcad 521 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  -> 
( x  C_  X  /\  E. y  e.  ( Ft  A ) y  C_  x ) ) )
36 elfg 17908 . . . . 5  |-  ( ( Ft  A )  e.  (
fBas `  X )  ->  ( x  e.  ( X filGen ( Ft  A ) )  <->  ( x  C_  X  /\  E. y  e.  ( Ft  A ) y  C_  x ) ) )
3717, 36syl 16 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  ( X
filGen ( Ft  A ) )  <->  ( x  C_  X  /\  E. y  e.  ( Ft  A ) y  C_  x ) ) )
3835, 37sylibrd 227 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  x  e.  ( X filGen ( Ft  A ) ) ) )
3938ssrdv 3356 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  F  C_  ( X filGen ( Ft  A ) ) )
4024, 39eqssd 3367 1  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen ( Ft  A ) )  =  F )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    \ cdif 3319    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   ` cfv 5457  (class class class)co 6084   ↾t crest 13653   fBascfbas 16694   filGencfg 16695   Filcfil 17882
This theorem is referenced by:  cfilres  19254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-rest 13655  df-fbas 16704  df-fg 16705  df-fil 17883
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