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Theorem elfilss 20107
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 504 . . 3  |-  ( A 
C_  X  ->  ( E. t  e.  F  t  C_  A  <->  ( A  C_  X  /\  E. t  e.  F  t  C_  A ) ) )
21adantl 466 . 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 20079 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
4 elfg 20102 . . . 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 465 . 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 20106 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
87eleq2d 2532 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
A  e.  F ) )
98adantr 465 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  ( A  e.  ( X filGen F )  <->  A  e.  F ) )
102, 6, 93bitr2rd 282 1  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  ( A  e.  F  <->  E. t  e.  F  t  C_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1762   E.wrex 2810    C_ wss 3471   ` cfv 5581  (class class class)co 6277   fBascfbas 18172   filGencfg 18173   Filcfil 20076
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 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-fbas 18182  df-fg 18183  df-fil 20077
This theorem is referenced by:  trfil3  20119
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