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Theorem elfilss 17587
Description: An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
elfilss  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  ( A  e.  F  <->  E. t  e.  F  t  C_  A ) )
Distinct variable groups:    t, F    t, X    t, A

Proof of Theorem elfilss
StepHypRef Expression
1 ibar 490 . . 3  |-  ( A 
C_  X  ->  ( E. t  e.  F  t  C_  A  <->  ( A  C_  X  /\  E. t  e.  F  t  C_  A ) ) )
21adantl 452 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  ( E. t  e.  F  t  C_  A  <->  ( A  C_  X  /\  E. t  e.  F  t  C_  A ) ) )
3 filfbas 17559 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
4 elfg 17582 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
( A  C_  X  /\  E. t  e.  F  t  C_  A ) ) )
53, 4syl 15 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
( A  C_  X  /\  E. t  e.  F  t  C_  A ) ) )
65adantr 451 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  ( A  e.  ( X filGen F )  <->  ( A  C_  X  /\  E. t  e.  F  t  C_  A ) ) )
7 fgfil 17586 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
87eleq2d 2363 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
A  e.  F ) )
98adantr 451 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  ( A  e.  ( X filGen F )  <->  A  e.  F ) )
102, 6, 93bitr2rd 273 1  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  ( A  e.  F  <->  E. t  e.  F  t  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   E.wrex 2557    C_ wss 3165   ` cfv 5271  (class class class)co 5874   fBascfbas 17534   filGencfg 17535   Filcfil 17556
This theorem is referenced by:  trfil3  17599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-fbas 17536  df-fg 17537  df-fil 17557
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