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Theorem trfilss 17600
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 13347 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  =  ran  ( x  e.  F  |->  ( x  i^i  A
) ) )
2 filin 17565 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F  /\  A  e.  F )  ->  (
x  i^i  A )  e.  F )
323expa 1151 . . . . 5  |-  ( ( ( F  e.  ( Fil `  X )  /\  x  e.  F
)  /\  A  e.  F )  ->  (
x  i^i  A )  e.  F )
43an32s 779 . . . 4  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  e.  F
)  /\  x  e.  F )  ->  (
x  i^i  A )  e.  F )
5 eqid 2296 . . . 4  |-  ( x  e.  F  |->  ( x  i^i  A ) )  =  ( x  e.  F  |->  ( x  i^i 
A ) )
64, 5fmptd 5700 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  (
x  e.  F  |->  ( x  i^i  A ) ) : F --> F )
7 frn 5411 . . 3  |-  ( ( x  e.  F  |->  ( x  i^i  A ) ) : F --> F  ->  ran  ( x  e.  F  |->  ( x  i^i  A
) )  C_  F
)
86, 7syl 15 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ran  ( x  e.  F  |->  ( x  i^i  A
) )  C_  F
)
91, 8eqsstrd 3225 1  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( Ft  A )  C_  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696    i^i cin 3164    C_ wss 3165    e. cmpt 4093   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Filcfil 17556
This theorem is referenced by:  fgtr  17601  flimrest  17694
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-reu 2563  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-iun 3923  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-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-rest 13343  df-fbas 17536  df-fil 17557
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