Theorem f1imacnv 5384

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 4852	. 2 |- ( ( `' F |` ( F " C ) ) " ( F " C ) ) = ( `' F " ( F " C ) )
2		df-f1 5128	. . . . . . 7 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
3	2	simprbi 273	. . . . . 6 |- ( F : A -1-1-> B -> Fun `' F )
4	3	adantr 274	. . . . 5 |- ( ( F : A -1-1-> B /\ C C_ A ) -> Fun `' F )
5		funcnvres 5196	. . . . 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 4880	. . 3 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' ( F |` C ) " ( F " C ) ) = ( ( `' F |` ( F " C ) ) " ( F " C ) ) )
8		f1ores 5382	. . . . 5 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
9		f1ocnv 5380	. . . . 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 4891	. . . . 5 |- ( `' ( F |` C ) " dom `' ( F |` C ) ) = ran `' ( F |` C )
12		f1odm 5371	. . . . . 6 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> dom `' ( F |` C ) = ( F " C ) )
13	12	imaeq2d 4881	. . . . 5 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> ( `' ( F |` C ) " dom `' ( F |` C ) ) = ( `' ( F |` C ) " ( F " C ) ) )
14		f1ofo 5374	. . . . . 6 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> `' ( F |` C ) : ( F " C ) -onto-> C )
15		forn 5348	. . . . . 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 2196	. . . 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 2174	. 2 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( ( `' F |` ( F " C ) ) " ( F " C ) ) = C )
20	1, 19	syl5eqr 2186	1 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' F " ( F " C ) ) = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   C_ wss 3071   `'ccnv 4538   dom cdm 4539   ran crn 4540   |` cres 4541   "cima 4542   Fun wfun 5117   -->wf 5119   -1-1->wf1 5120   -onto->wfo 5121   -1-1-onto->wf1o 5122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130

This theorem is referenced by:  f1opw2  5976  ssenen  6745  hmeoopn  12490  hmeocld  12491  hmeontr  12492