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Theorem fmid 20193
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 20081 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 f1oi 5849 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1ofo 5821 . . . . 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 2467 . . . . 5  |-  ( X
filGen F )  =  ( X filGen F )
65elfm3 20183 . . . 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 20108 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
98rexeqdv 3065 . . 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 20085 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  s  C_  X )
11 resiima 5349 . . . . . . . 8  |-  ( s 
C_  X  ->  (
(  _I  |`  X )
" s )  =  s )
1210, 11syl 16 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
(  _I  |`  X )
" s )  =  s )
1312eqeq2d 2481 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
t  =  ( (  _I  |`  X ) " s )  <->  t  =  s ) )
14 equcom 1743 . . . . . 6  |-  ( s  =  t  <->  t  =  s )
1513, 14syl6bbr 263 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
t  =  ( (  _I  |`  X ) " s )  <->  s  =  t ) )
1615rexbidva 2970 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( E. s  e.  F  t  =  ( (  _I  |`  X ) " s
)  <->  E. s  e.  F  s  =  t )
)
17 risset 2987 . . . 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 2464 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 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476    _I cid 4790    |` cres 5001   "cima 5002   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   fBascfbas 18174   filGencfg 18175   Filcfil 20078    FilMap cfm 20166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-fbas 18184  df-fg 18185  df-fil 20079  df-fm 20171
This theorem is referenced by:  ufldom  20195
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