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Theorem fgtr 17914
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 17872 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 fbncp 17863 . . . . . . . 8  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  ( X  \  A )  e.  F )
31, 2sylan 458 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  -.  ( X  \  A )  e.  F )
4 filelss 17876 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  C_  X )
5 trfil3 17912 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  (
( Ft  A )  e.  ( Fil `  A )  <->  -.  ( X  \  A
)  e.  F ) )
64, 5syldan 457 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
( Ft  A )  e.  ( Fil `  A )  <->  -.  ( X  \  A
)  e.  F ) )
73, 6mpbird 224 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  e.  ( Fil `  A ) )
8 filfbas 17872 . . . . . 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 13651 . . . . . 6  |-  ( Ft  A )  C_  ~P A
11 sspwb 4405 . . . . . . 7  |-  ( A 
C_  X  <->  ~P A  C_ 
~P X )
124, 11sylib 189 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ~P A  C_  ~P X )
1310, 12syl5ss 3351 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  C_  ~P X )
14 filtop 17879 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
1514adantr 452 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  X  e.  F )
16 fbasweak 17889 . . . . 5  |-  ( ( ( Ft  A )  e.  (
fBas `  A )  /\  ( Ft  A )  C_  ~P X  /\  X  e.  F
)  ->  ( Ft  A
)  e.  ( fBas `  X ) )
179, 13, 15, 16syl3anc 1184 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  e.  (
fBas `  X )
)
181adantr 452 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  F  e.  ( fBas `  X
) )
19 trfilss 17913 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  C_  F
)
20 fgss 17897 . . . 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 1184 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen ( Ft  A ) )  C_  ( X filGen F ) )
22 fgfil 17899 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
2322adantr 452 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen F )  =  F )
2421, 23sseqtrd 3376 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen ( Ft  A ) )  C_  F )
25 filelss 17876 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F )  ->  x  C_  X )
2625ex 424 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  x  C_  X ) )
2726adantr 452 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  x  C_  X ) )
28 elrestr 13648 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  x  e.  F )  ->  (
x  i^i  A )  e.  ( Ft  A ) )
29283expa 1153 . . . . . . 7  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  e.  F
)  /\  x  e.  F )  ->  (
x  i^i  A )  e.  ( Ft  A ) )
30 inss1 3553 . . . . . . 7  |-  ( x  i^i  A )  C_  x
31 sseq1 3361 . . . . . . . 8  |-  ( y  =  ( x  i^i 
A )  ->  (
y  C_  x  <->  ( x  i^i  A )  C_  x
) )
3231rspcev 3044 . . . . . . 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 644 . . . . . 6  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  e.  F
)  /\  x  e.  F )  ->  E. y  e.  ( Ft  A ) y  C_  x )
3433ex 424 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  E. y  e.  ( Ft  A ) y  C_  x ) )
3527, 34jcad 520 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  -> 
( x  C_  X  /\  E. y  e.  ( Ft  A ) y  C_  x ) ) )
36 elfg 17895 . . . . 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 226 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  x  e.  ( X filGen ( Ft  A ) ) ) )
3938ssrdv 3346 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  F  C_  ( X filGen ( Ft  A ) ) )
4024, 39eqssd 3357 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698    \ cdif 3309    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   ` cfv 5446  (class class class)co 6073   ↾t crest 13640   fBascfbas 16681   filGencfg 16682   Filcfil 17869
This theorem is referenced by:  cfilres  19241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-rest 13642  df-fbas 16691  df-fg 16692  df-fil 17870
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