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Theorem fgtr 20264
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 20222 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 fbncp 20213 . . . . . . . 8  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  ( X  \  A )  e.  F )
31, 2sylan 471 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  -.  ( X  \  A )  e.  F )
4 filelss 20226 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  C_  X )
5 trfil3 20262 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  (
( Ft  A )  e.  ( Fil `  A )  <->  -.  ( X  \  A
)  e.  F ) )
64, 5syldan 470 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
( Ft  A )  e.  ( Fil `  A )  <->  -.  ( X  \  A
)  e.  F ) )
73, 6mpbird 232 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  e.  ( Fil `  A ) )
8 filfbas 20222 . . . . . 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 14706 . . . . . 6  |-  ( Ft  A )  C_  ~P A
11 sspwb 4686 . . . . . . 7  |-  ( A 
C_  X  <->  ~P A  C_ 
~P X )
124, 11sylib 196 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ~P A  C_  ~P X )
1310, 12syl5ss 3500 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  C_  ~P X )
14 filtop 20229 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
1514adantr 465 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  X  e.  F )
16 fbasweak 20239 . . . . 5  |-  ( ( ( Ft  A )  e.  (
fBas `  A )  /\  ( Ft  A )  C_  ~P X  /\  X  e.  F
)  ->  ( Ft  A
)  e.  ( fBas `  X ) )
179, 13, 15, 16syl3anc 1229 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  e.  (
fBas `  X )
)
181adantr 465 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  F  e.  ( fBas `  X
) )
19 trfilss 20263 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  C_  F
)
20 fgss 20247 . . . 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 1229 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen ( Ft  A ) )  C_  ( X filGen F ) )
22 fgfil 20249 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
2322adantr 465 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen F )  =  F )
2421, 23sseqtrd 3525 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen ( Ft  A ) )  C_  F )
25 filelss 20226 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F )  ->  x  C_  X )
2625ex 434 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  x  C_  X ) )
2726adantr 465 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  x  C_  X ) )
28 elrestr 14703 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  x  e.  F )  ->  (
x  i^i  A )  e.  ( Ft  A ) )
29283expa 1197 . . . . . . 7  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  e.  F
)  /\  x  e.  F )  ->  (
x  i^i  A )  e.  ( Ft  A ) )
30 inss1 3703 . . . . . . 7  |-  ( x  i^i  A )  C_  x
31 sseq1 3510 . . . . . . . 8  |-  ( y  =  ( x  i^i 
A )  ->  (
y  C_  x  <->  ( x  i^i  A )  C_  x
) )
3231rspcev 3196 . . . . . . 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 662 . . . . . 6  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  e.  F
)  /\  x  e.  F )  ->  E. y  e.  ( Ft  A ) y  C_  x )
3433ex 434 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  E. y  e.  ( Ft  A ) y  C_  x ) )
3527, 34jcad 533 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  -> 
( x  C_  X  /\  E. y  e.  ( Ft  A ) y  C_  x ) ) )
36 elfg 20245 . . . . 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 234 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  x  e.  ( X filGen ( Ft  A ) ) ) )
3938ssrdv 3495 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  F  C_  ( X filGen ( Ft  A ) ) )
4024, 39eqssd 3506 1  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen ( Ft  A ) )  =  F )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   E.wrex 2794    \ cdif 3458    i^i cin 3460    C_ wss 3461   ~Pcpw 3997   ` cfv 5578  (class class class)co 6281   ↾t crest 14695   fBascfbas 18280   filGencfg 18281   Filcfil 20219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-rest 14697  df-fbas 18290  df-fg 18291  df-fil 20220
This theorem is referenced by:  cfilres  21608
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