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Theorem fmid 19533
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 19421 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 f1oi 5676 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1ofo 5648 . . . . 5  |-  ( (  _I  |`  X ) : X -1-1-onto-> X  ->  (  _I  |`  X ) : X -onto-> X )
42, 3ax-mp 5 . . . 4  |-  (  _I  |`  X ) : X -onto-> X
5 eqid 2443 . . . . 5  |-  ( X
filGen F )  =  ( X filGen F )
65elfm3 19523 . . . 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 662 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( ( X  FilMap  (  _I  |`  X )
) `  F )  <->  E. s  e.  ( X
filGen F ) t  =  ( (  _I  |`  X )
" s ) ) )
8 fgfil 19448 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
98rexeqdv 2924 . . 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 19425 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  s  C_  X )
11 resiima 5183 . . . . . . . 8  |-  ( s 
C_  X  ->  (
(  _I  |`  X )
" s )  =  s )
1210, 11syl 16 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
(  _I  |`  X )
" s )  =  s )
1312eqeq2d 2454 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
t  =  ( (  _I  |`  X ) " s )  <->  t  =  s ) )
14 equcom 1732 . . . . . 6  |-  ( s  =  t  <->  t  =  s )
1513, 14syl6bbr 263 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
t  =  ( (  _I  |`  X ) " s )  <->  s  =  t ) )
1615rexbidva 2732 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( E. s  e.  F  t  =  ( (  _I  |`  X ) " s
)  <->  E. s  e.  F  s  =  t )
)
17 risset 2763 . . . 4  |-  ( t  e.  F  <->  E. s  e.  F  s  =  t )
1816, 17syl6bbr 263 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( E. s  e.  F  t  =  ( (  _I  |`  X ) " s
)  <->  t  e.  F
) )
197, 9, 183bitrd 279 . 2  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( ( X  FilMap  (  _I  |`  X )
) `  F )  <->  t  e.  F ) )
2019eqrdv 2441 1  |-  ( F  e.  ( Fil `  X
)  ->  ( ( X  FilMap  (  _I  |`  X ) ) `  F )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716    C_ wss 3328    _I cid 4631    |` cres 4842   "cima 4843   -onto->wfo 5416   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091   fBascfbas 17804   filGencfg 17805   Filcfil 19418    FilMap cfm 19506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-fbas 17814  df-fg 17815  df-fil 19419  df-fm 19511
This theorem is referenced by:  ufldom  19535
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