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Theorem trfil1 20115
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 3696 . . . . 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 20084 . . . . . 6  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
7 ssexg 4586 . . . . . 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 14673 . . . . 5  |-  ( ( L  e.  ( Fil `  Y )  /\  A  e.  _V  /\  Y  e.  L )  ->  ( Y  i^i  A )  e.  ( Lt  A ) )
114, 8, 9, 10syl3anc 1223 . . . 4  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  ( Y  i^i  A )  e.  ( Lt  A ) )
123, 11eqeltrrd 2549 . . 3  |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  A  e.  ( Lt  A ) )
13 elssuni 4268 . . 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 14676 . . . 4  |-  ( Lt  A )  C_  ~P A
16 sspwuni 4404 . . . 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 3514 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 1374    e. wcel 1762   _Vcvv 3106    i^i cin 3468    C_ wss 3469   ~Pcpw 4003   U.cuni 4238   ` cfv 5579  (class class class)co 6275   ↾t crest 14665   Filcfil 20074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-rest 14667  df-fbas 18180  df-fil 20075
This theorem is referenced by: (None)
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