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Theorem trfil1 19577
Description: Conditions for the trace of a filter  L to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
trfil1  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  A  =  U. ( Lt  A ) )

Proof of Theorem trfil1
StepHypRef Expression
1 simpr 461 . . . . 5  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  A  C_  Y )
2 dfss1 3655 . . . . 5  |-  ( A 
C_  Y  <->  ( Y  i^i  A )  =  A )
31, 2sylib 196 . . . 4  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  ( Y  i^i  A )  =  A )
4 simpl 457 . . . . 5  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  L  e.  ( Fil `  Y
) )
5 id 22 . . . . . 6  |-  ( A 
C_  Y  ->  A  C_  Y )
6 filtop 19546 . . . . . 6  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
7 ssexg 4538 . . . . . 6  |-  ( ( A  C_  Y  /\  Y  e.  L )  ->  A  e.  _V )
85, 6, 7syl2anr 478 . . . . 5  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  A  e.  _V )
96adantr 465 . . . . 5  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  Y  e.  L )
10 elrestr 14471 . . . . 5  |-  ( ( L  e.  ( Fil `  Y )  /\  A  e.  _V  /\  Y  e.  L )  ->  ( Y  i^i  A )  e.  ( Lt  A ) )
114, 8, 9, 10syl3anc 1219 . . . 4  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  ( Y  i^i  A )  e.  ( Lt  A ) )
123, 11eqeltrrd 2540 . . 3  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  A  e.  ( Lt  A ) )
13 elssuni 4221 . . 3  |-  ( A  e.  ( Lt  A )  ->  A  C_  U. ( Lt  A ) )
1412, 13syl 16 . 2  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  A  C_ 
U. ( Lt  A ) )
15 restsspw 14474 . . . 4  |-  ( Lt  A )  C_  ~P A
16 sspwuni 4356 . . . 4  |-  ( ( Lt  A )  C_  ~P A 
<-> 
U. ( Lt  A ) 
C_  A )
1715, 16mpbi 208 . . 3  |-  U. ( Lt  A )  C_  A
1817a1i 11 . 2  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  U. ( Lt  A )  C_  A
)
1914, 18eqssd 3473 1  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  A  =  U. ( Lt  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3070    i^i cin 3427    C_ wss 3428   ~Pcpw 3960   U.cuni 4191   ` cfv 5518  (class class class)co 6192   ↾t crest 14463   Filcfil 19536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-rest 14465  df-fbas 17925  df-fil 19537
This theorem is referenced by: (None)
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