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Theorem fgtr 17585
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 17543 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 fbncp 17534 . . . . . . . 8  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F )  ->  -.  ( X  \  A )  e.  F )
31, 2sylan 457 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  -.  ( X  \  A )  e.  F )
4 filelss 17547 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  C_  X )
5 trfil3 17583 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  (
( Ft  A )  e.  ( Fil `  A )  <->  -.  ( X  \  A
)  e.  F ) )
64, 5syldan 456 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
( Ft  A )  e.  ( Fil `  A )  <->  -.  ( X  \  A
)  e.  F ) )
73, 6mpbird 223 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  e.  ( Fil `  A ) )
8 filfbas 17543 . . . . . 6  |-  ( ( Ft  A )  e.  ( Fil `  A )  ->  ( Ft  A )  e.  ( fBas `  A
) )
97, 8syl 15 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  e.  (
fBas `  A )
)
10 restsspw 13336 . . . . . 6  |-  ( Ft  A )  C_  ~P A
11 sspwb 4223 . . . . . . 7  |-  ( A 
C_  X  <->  ~P A  C_ 
~P X )
124, 11sylib 188 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ~P A  C_  ~P X )
1310, 12syl5ss 3190 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  C_  ~P X )
14 filtop 17550 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
1514adantr 451 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  X  e.  F )
16 fbasweak 17560 . . . . 5  |-  ( ( ( Ft  A )  e.  (
fBas `  A )  /\  ( Ft  A )  C_  ~P X  /\  X  e.  F
)  ->  ( Ft  A
)  e.  ( fBas `  X ) )
179, 13, 15, 16syl3anc 1182 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  e.  (
fBas `  X )
)
181adantr 451 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  F  e.  ( fBas `  X
) )
19 trfilss 17584 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  C_  F
)
20 fgss 17568 . . . 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 1182 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen ( Ft  A ) )  C_  ( X filGen F ) )
22 fgfil 17570 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
2322adantr 451 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen F )  =  F )
2421, 23sseqtrd 3214 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( X filGen ( Ft  A ) )  C_  F )
25 filelss 17547 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F )  ->  x  C_  X )
2625ex 423 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  x  C_  X ) )
2726adantr 451 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  x  C_  X ) )
28 elrestr 13333 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  x  e.  F )  ->  (
x  i^i  A )  e.  ( Ft  A ) )
29283expa 1151 . . . . . . 7  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  e.  F
)  /\  x  e.  F )  ->  (
x  i^i  A )  e.  ( Ft  A ) )
30 inss1 3389 . . . . . . 7  |-  ( x  i^i  A )  C_  x
31 sseq1 3199 . . . . . . . 8  |-  ( y  =  ( x  i^i 
A )  ->  (
y  C_  x  <->  ( x  i^i  A )  C_  x
) )
3231rspcev 2884 . . . . . . 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 643 . . . . . 6  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  e.  F
)  /\  x  e.  F )  ->  E. y  e.  ( Ft  A ) y  C_  x )
3433ex 423 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  E. y  e.  ( Ft  A ) y  C_  x ) )
3527, 34jcad 519 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  -> 
( x  C_  X  /\  E. y  e.  ( Ft  A ) y  C_  x ) ) )
36 elfg 17566 . . . . 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 15 . . . 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 225 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  ->  x  e.  ( X filGen ( Ft  A ) ) ) )
3938ssrdv 3185 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  F  C_  ( X filGen ( Ft  A ) ) )
4024, 39eqssd 3196 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   fBascfbas 17518   filGencfg 17519   Filcfil 17540
This theorem is referenced by:  cfilres  18722
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-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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-1st 6122  df-2nd 6123  df-rest 13327  df-fbas 17520  df-fg 17521  df-fil 17541
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