Theorem f1imacnv 5600

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 5046	. 2 |- ( ( `' F |` ( F " C ) ) " ( F " C ) ) = ( `' F " ( F " C ) )
2		df-f1 5331	. . . . . . 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 5403	. . . . 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 5075	. . 3 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' ( F |` C ) " ( F " C ) ) = ( ( `' F |` ( F " C ) ) " ( F " C ) ) )
8		f1ores 5598	. . . . 5 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
9		f1ocnv 5596	. . . . 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 5086	. . . . 5 |- ( `' ( F |` C ) " dom `' ( F |` C ) ) = ran `' ( F |` C )
12		f1odm 5587	. . . . . 6 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> dom `' ( F |` C ) = ( F " C ) )
13	12	imaeq2d 5076	. . . . 5 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> ( `' ( F |` C ) " dom `' ( F |` C ) ) = ( `' ( F |` C ) " ( F " C ) ) )
14		f1ofo 5590	. . . . . 6 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> `' ( F |` C ) : ( F " C ) -onto-> C )
15		forn 5562	. . . . . 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 2288	. . . 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 2266	. 2 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( ( `' F |` ( F " C ) ) " ( F " C ) ) = C )
20	1, 19	eqtr3id 2278	1 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' F " ( F " C ) ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   C_ wss 3200   `'ccnv 4724   dom cdm 4725   ran crn 4726   |` cres 4727   "cima 4728   Fun wfun 5320   -->wf 5322   -1-1->wf1 5323   -onto->wfo 5324   -1-1-onto->wf1o 5325

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333

This theorem is referenced by:  f1opw2  6228  ssenen  7036  hmeoopn  15034  hmeocld  15035  hmeontr  15036