Theorem f1imacnv 5518

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 4976	. 2 |- ( ( `' F |` ( F " C ) ) " ( F " C ) ) = ( `' F " ( F " C ) )
2		df-f1 5260	. . . . . . 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 5328	. . . . 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 5005	. . 3 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' ( F |` C ) " ( F " C ) ) = ( ( `' F |` ( F " C ) ) " ( F " C ) ) )
8		f1ores 5516	. . . . 5 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
9		f1ocnv 5514	. . . . 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 5016	. . . . 5 |- ( `' ( F |` C ) " dom `' ( F |` C ) ) = ran `' ( F |` C )
12		f1odm 5505	. . . . . 6 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> dom `' ( F |` C ) = ( F " C ) )
13	12	imaeq2d 5006	. . . . 5 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> ( `' ( F |` C ) " dom `' ( F |` C ) ) = ( `' ( F |` C ) " ( F " C ) ) )
14		f1ofo 5508	. . . . . 6 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> `' ( F |` C ) : ( F " C ) -onto-> C )
15		forn 5480	. . . . . 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 2250	. . . 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 2228	. 2 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( ( `' F |` ( F " C ) ) " ( F " C ) ) = C )
20	1, 19	eqtr3id 2240	1 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' F " ( F " C ) ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   C_ wss 3154   `'ccnv 4659   dom cdm 4660   ran crn 4661   |` cres 4662   "cima 4663   Fun wfun 5249   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262

This theorem is referenced by:  f1opw2  6126  ssenen  6909  hmeoopn  14490  hmeocld  14491  hmeontr  14492