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Theorem fmid 20439
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 20327 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 f1oi 5841 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1ofo 5813 . . . . 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 20429 . . . 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 20354 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
98rexeqdv 3047 . . 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 20331 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  s  C_  X )
11 resiima 5341 . . . . . . . 8  |-  ( s 
C_  X  ->  (
(  _I  |`  X )
" s )  =  s )
1210, 11syl 16 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
(  _I  |`  X )
" s )  =  s )
1312eqeq2d 2457 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
t  =  ( (  _I  |`  X ) " s )  <->  t  =  s ) )
14 equcom 1780 . . . . . 6  |-  ( s  =  t  <->  t  =  s )
1513, 14syl6bbr 263 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  s  e.  F )  ->  (
t  =  ( (  _I  |`  X ) " s )  <->  s  =  t ) )
1615rexbidva 2951 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( E. s  e.  F  t  =  ( (  _I  |`  X ) " s
)  <->  E. s  e.  F  s  =  t )
)
17 risset 2968 . . . 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 2440 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 1383    e. wcel 1804   E.wrex 2794    C_ wss 3461    _I cid 4780    |` cres 4991   "cima 4992   -onto->wfo 5576   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   fBascfbas 18385   filGencfg 18386   Filcfil 20324    FilMap cfm 20412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-fbas 18395  df-fg 18396  df-fil 20325  df-fm 20417
This theorem is referenced by:  ufldom  20441
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