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Theorem elfilss 17822
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 491 . . 3  |-  ( A 
C_  X  ->  ( E. t  e.  F  t  C_  A  <->  ( A  C_  X  /\  E. t  e.  F  t  C_  A ) ) )
21adantl 453 . 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 17794 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
4 elfg 17817 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
( A  C_  X  /\  E. t  e.  F  t  C_  A ) ) )
53, 4syl 16 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
( A  C_  X  /\  E. t  e.  F  t  C_  A ) ) )
65adantr 452 . 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 17821 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
87eleq2d 2447 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
A  e.  F ) )
98adantr 452 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  ( A  e.  ( X filGen F )  <->  A  e.  F ) )
102, 6, 93bitr2rd 274 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 177    /\ wa 359    e. wcel 1717   E.wrex 2643    C_ wss 3256   ` cfv 5387  (class class class)co 6013   fBascfbas 16608   filGencfg 16609   Filcfil 17791
This theorem is referenced by:  trfil3  17834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-fbas 16616  df-fg 16617  df-fil 17792
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