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Theorem fmid 17915
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 17803 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 f1oi 5655 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1ofo 5623 . . . . 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 2389 . . . . 5  |-  ( X
filGen F )  =  ( X filGen F )
65elfm3 17905 . . . 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 644 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( ( X  FilMap  (  _I  |`  X )
) `  F )  <->  E. s  e.  ( X
filGen F ) t  =  ( (  _I  |`  X )
" s ) ) )
8 fgfil 17830 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
98rexeqdv 2856 . . 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 17807 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  s  C_  X )
11 resiima 5162 . . . . . . . 8  |-  ( s 
C_  X  ->  (
(  _I  |`  X )
" s )  =  s )
1210, 11syl 16 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
(  _I  |`  X )
" s )  =  s )
1312eqeq2d 2400 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
t  =  ( (  _I  |`  X ) " s )  <->  t  =  s ) )
14 equcom 1687 . . . . . 6  |-  ( s  =  t  <->  t  =  s )
1513, 14syl6bbr 255 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
t  =  ( (  _I  |`  X ) " s )  <->  s  =  t ) )
1615rexbidva 2668 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( E. s  e.  F  t  =  ( (  _I  |`  X ) " s
)  <->  E. s  e.  F  s  =  t )
)
17 risset 2698 . . . 4  |-  ( t  e.  F  <->  E. s  e.  F  s  =  t )
1816, 17syl6bbr 255 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( E. s  e.  F  t  =  ( (  _I  |`  X ) " s
)  <->  t  e.  F
) )
197, 9, 183bitrd 271 . 2  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( ( X  FilMap  (  _I  |`  X )
) `  F )  <->  t  e.  F ) )
2019eqrdv 2387 1  |-  ( F  e.  ( Fil `  X
)  ->  ( ( X  FilMap  (  _I  |`  X ) ) `  F )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2652    C_ wss 3265    _I cid 4436    |` cres 4822   "cima 4823   -onto->wfo 5394   -1-1-onto->wf1o 5395   ` cfv 5396  (class class class)co 6022   fBascfbas 16617   filGencfg 16618   Filcfil 17800    FilMap cfm 17888
This theorem is referenced by:  ufldom  17917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-fbas 16625  df-fg 16626  df-fil 17801  df-fm 17893
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