Theorem f1imacnv 5589

Description: Preimage of an image. (Contributed by NM, 30-Sep-2004.)

Assertion

Ref	Expression
f1imacnv	|- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' F " ( F " C ) ) = C )

Proof of Theorem f1imacnv

Step	Hyp	Ref	Expression
1		resima 5038	. 2 |- ( ( `' F |` ( F " C ) ) " ( F " C ) ) = ( `' F " ( F " C ) )
2		df-f1 5323	. . . . . . 7 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
3	2	simprbi 275	. . . . . 6 |- ( F : A -1-1-> B -> Fun `' F )
4	3	adantr 276	. . . . 5 |- ( ( F : A -1-1-> B /\ C C_ A ) -> Fun `' F )
5		funcnvres 5394	. . . . 5 |- ( Fun `' F -> `' ( F |` C ) = ( `' F |` ( F " C ) ) )
6	4, 5	syl 14	. . . 4 |- ( ( F : A -1-1-> B /\ C C_ A ) -> `' ( F |` C ) = ( `' F |` ( F " C ) ) )
7	6	imaeq1d 5067	. . 3 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' ( F |` C ) " ( F " C ) ) = ( ( `' F |` ( F " C ) ) " ( F " C ) ) )
8		f1ores 5587	. . . . 5 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
9		f1ocnv 5585	. . . . 5 |- ( ( F |` C ) : C -1-1-onto-> ( F " C ) -> `' ( F |` C ) : ( F " C ) -1-1-onto-> C )
10	8, 9	syl 14	. . . 4 |- ( ( F : A -1-1-> B /\ C C_ A ) -> `' ( F |` C ) : ( F " C ) -1-1-onto-> C )
11		imadmrn 5078	. . . . 5 |- ( `' ( F |` C ) " dom `' ( F |` C ) ) = ran `' ( F |` C )
12		f1odm 5576	. . . . . 6 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> dom `' ( F |` C ) = ( F " C ) )
13	12	imaeq2d 5068	. . . . 5 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> ( `' ( F |` C ) " dom `' ( F |` C ) ) = ( `' ( F |` C ) " ( F " C ) ) )
14		f1ofo 5579	. . . . . 6 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> `' ( F |` C ) : ( F " C ) -onto-> C )
15		forn 5551	. . . . . 6 |- ( `' ( F |` C ) : ( F " C ) -onto-> C -> ran `' ( F |` C ) = C )
16	14, 15	syl 14	. . . . 5 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> ran `' ( F |` C ) = C )
17	11, 13, 16	3eqtr3a 2286	. . . 4 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> ( `' ( F |` C ) " ( F " C ) ) = C )
18	10, 17	syl 14	. . 3 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' ( F |` C ) " ( F " C ) ) = C )
19	7, 18	eqtr3d 2264	. 2 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( ( `' F |` ( F " C ) ) " ( F " C ) ) = C )
20	1, 19	eqtr3id 2276	1 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' F " ( F " C ) ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   C_ wss 3197   `'ccnv 4718   dom cdm 4719   ran crn 4720   |` cres 4721   "cima 4722   Fun wfun 5312   -->wf 5314   -1-1->wf1 5315   -onto->wfo 5316   -1-1-onto->wf1o 5317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325

This theorem is referenced by:  f1opw2  6212  ssenen  7012  hmeoopn  14985  hmeocld  14986  hmeontr  14987