Theorem fmid 21764

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.

Step	Hyp	Ref	Expression
1		filfbas 21652	. . . 4 |- ( F e. ( Fil ` X ) -> F e. ( fBas ` X ) )
2		f1oi 6174	. . . . 5 |- ( _I |` X ) : X -1-1-onto-> X
3		f1ofo 6144	. . . . 5 |- ( ( _I |` X ) : X -1-1-onto-> X -> ( _I |` X ) : X -onto-> X )
4	2, 3	ax-mp 5	. . . 4 |- ( _I |` X ) : X -onto-> X
5		eqid 2622	. . . . 5 |- ( X filGen F ) = ( X filGen F )
6	5	elfm3 21754	. . . 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 ) ) )
7	1, 4, 6	sylancl 694	. . 3 |- ( F e. ( Fil ` X ) -> ( t e. ( ( X FilMap ( _I |` X ) ) ` F ) <-> E. s e. ( X filGen F ) t = ( ( _I |` X ) " s ) ) )
8		fgfil 21679	. . . 4 |- ( F e. ( Fil ` X ) -> ( X filGen F ) = F )
9	8	rexeqdv 3145	. . 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 21656	. . . . . . . 8 |- ( ( F e. ( Fil ` X ) /\ s e. F ) -> s C_ X )
11		resiima 5480	. . . . . . . 8 |- ( s C_ X -> ( ( _I |` X ) " s ) = s )
12	10, 11	syl 17	. . . . . . 7 |- ( ( F e. ( Fil ` X ) /\ s e. F ) -> ( ( _I |` X ) " s ) = s )
13	12	eqeq2d 2632	. . . . . 6 |- ( ( F e. ( Fil ` X ) /\ s e. F ) -> ( t = ( ( _I |` X ) " s ) <-> t = s ) )
14		equcom 1945	. . . . . 6 |- ( s = t <-> t = s )
15	13, 14	syl6bbr 278	. . . . 5 |- ( ( F e. ( Fil ` X ) /\ s e. F ) -> ( t = ( ( _I |` X ) " s ) <-> s = t ) )
16	15	rexbidva 3049	. . . 4 |- ( F e. ( Fil ` X ) -> ( E. s e. F t = ( ( _I |` X ) " s ) <-> E. s e. F s = t ) )
17		risset 3062	. . . 4 |- ( t e. F <-> E. s e. F s = t )
18	16, 17	syl6bbr 278	. . 3 |- ( F e. ( Fil ` X ) -> ( E. s e. F t = ( ( _I |` X ) " s ) <-> t e. F ) )
19	7, 9, 18	3bitrd 294	. 2 |- ( F e. ( Fil ` X ) -> ( t e. ( ( X FilMap ( _I |` X ) ) ` F ) <-> t e. F ) )
20	19	eqrdv 2620	1 |- ( F e. ( Fil ` X ) -> ( ( X FilMap ( _I |` X ) ) ` F ) = F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   _I cid 5023   |` cres 5116   "cima 5117   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   fBascfbas 19734   filGencfg 19735   Filcfil 21649   FilMap cfm 21737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-fg 19744  df-fil 21650  df-fm 21742

This theorem is referenced by:  ufldom  21766