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Theorem fmid 17671
Description: The filter map applied to the identity. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Mario Carneiro, 27-Aug-2015.)
Assertion
Ref Expression
fmid  |-  ( F  e.  ( Fil `  X
)  ->  ( ( X  FilMap  (  _I  |`  X ) ) `  F )  =  F )

Proof of Theorem fmid
Dummy variables  t 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filfbas 17559 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 f1oi 5527 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1ofo 5495 . . . . 5  |-  ( (  _I  |`  X ) : X -1-1-onto-> X  ->  (  _I  |`  X ) : X -onto-> X )
42, 3ax-mp 8 . . . 4  |-  (  _I  |`  X ) : X -onto-> X
5 eqid 2296 . . . . 5  |-  ( X
filGen F )  =  ( X filGen F )
65elfm3 17661 . . . 4  |-  ( ( F  e.  ( fBas `  X )  /\  (  _I  |`  X ) : X -onto-> X )  ->  (
t  e.  ( ( X  FilMap  (  _I  |`  X ) ) `  F )  <->  E. s  e.  ( X filGen F ) t  =  ( (  _I  |`  X ) " s
) ) )
71, 4, 6sylancl 643 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( ( X  FilMap  (  _I  |`  X )
) `  F )  <->  E. s  e.  ( X
filGen F ) t  =  ( (  _I  |`  X )
" s ) ) )
8 fgfil 17586 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
98rexeqdv 2756 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( E. s  e.  ( X filGen F ) t  =  ( (  _I  |`  X )
" s )  <->  E. s  e.  F  t  =  ( (  _I  |`  X )
" s ) ) )
10 filelss 17563 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  s  C_  X )
11 resiima 5045 . . . . . . . 8  |-  ( s 
C_  X  ->  (
(  _I  |`  X )
" s )  =  s )
1210, 11syl 15 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
(  _I  |`  X )
" s )  =  s )
1312eqeq2d 2307 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
t  =  ( (  _I  |`  X ) " s )  <->  t  =  s ) )
14 equcom 1665 . . . . . 6  |-  ( s  =  t  <->  t  =  s )
1513, 14syl6bbr 254 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
t  =  ( (  _I  |`  X ) " s )  <->  s  =  t ) )
1615rexbidva 2573 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( E. s  e.  F  t  =  ( (  _I  |`  X ) " s
)  <->  E. s  e.  F  s  =  t )
)
17 risset 2603 . . . 4  |-  ( t  e.  F  <->  E. s  e.  F  s  =  t )
1816, 17syl6bbr 254 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( E. s  e.  F  t  =  ( (  _I  |`  X ) " s
)  <->  t  e.  F
) )
197, 9, 183bitrd 270 . 2  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( ( X  FilMap  (  _I  |`  X )
) `  F )  <->  t  e.  F ) )
2019eqrdv 2294 1  |-  ( F  e.  ( Fil `  X
)  ->  ( ( X  FilMap  (  _I  |`  X ) ) `  F )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165    _I cid 4320    |` cres 4707   "cima 4708   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   fBascfbas 17534   filGencfg 17535   Filcfil 17556    FilMap cfm 17644
This theorem is referenced by:  ufldom  17673
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-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649
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