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Theorem trfilss 17921
Description: If  A is a member of the filter, then the filter truncated to  A is a subset of the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
trfilss  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  C_  F
)

Proof of Theorem trfilss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 restval 13654 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  =  ran  ( x  e.  F  |->  ( x  i^i  A
) ) )
2 filin 17886 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F  /\  A  e.  F )  ->  (
x  i^i  A )  e.  F )
323expa 1153 . . . . 5  |-  ( ( ( F  e.  ( Fil `  X )  /\  x  e.  F
)  /\  A  e.  F )  ->  (
x  i^i  A )  e.  F )
43an32s 780 . . . 4  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  e.  F
)  /\  x  e.  F )  ->  (
x  i^i  A )  e.  F )
5 eqid 2436 . . . 4  |-  ( x  e.  F  |->  ( x  i^i  A ) )  =  ( x  e.  F  |->  ( x  i^i 
A ) )
64, 5fmptd 5893 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  |->  ( x  i^i  A ) ) : F --> F )
7 frn 5597 . . 3  |-  ( ( x  e.  F  |->  ( x  i^i  A ) ) : F --> F  ->  ran  ( x  e.  F  |->  ( x  i^i  A
) )  C_  F
)
86, 7syl 16 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ran  ( x  e.  F  |->  ( x  i^i  A
) )  C_  F
)
91, 8eqsstrd 3382 1  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  C_  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    i^i cin 3319    C_ wss 3320    e. cmpt 4266   ran crn 4879   -->wf 5450   ` cfv 5454  (class class class)co 6081   ↾t crest 13648   Filcfil 17877
This theorem is referenced by:  fgtr  17922  flimrest  18015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-rest 13650  df-fbas 16699  df-fil 17878
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