Theorem f1imacnv 5636

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 5076	. 2 |- ( ( `' F |` ( F " C ) ) " ( F " C ) ) = ( `' F " ( F " C ) )
2		df-f1 5362	. . . . . . 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 5434	. . . . 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 5105	. . 3 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' ( F |` C ) " ( F " C ) ) = ( ( `' F |` ( F " C ) ) " ( F " C ) ) )
8		f1ores 5634	. . . . 5 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
9		f1ocnv 5632	. . . . 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 5116	. . . . 5 |- ( `' ( F |` C ) " dom `' ( F |` C ) ) = ran `' ( F |` C )
12		f1odm 5623	. . . . . 6 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> dom `' ( F |` C ) = ( F " C ) )
13	12	imaeq2d 5106	. . . . 5 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> ( `' ( F |` C ) " dom `' ( F |` C ) ) = ( `' ( F |` C ) " ( F " C ) ) )
14		f1ofo 5626	. . . . . 6 |- ( `' ( F |` C ) : ( F " C ) -1-1-onto-> C -> `' ( F |` C ) : ( F " C ) -onto-> C )
15		forn 5598	. . . . . 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 2291	. . . 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 2269	. 2 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( ( `' F |` ( F " C ) ) " ( F " C ) ) = C )
20	1, 19	eqtr3id 2281	1 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' F " ( F " C ) ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   C_ wss 3214   `'ccnv 4753   dom cdm 4754   ran crn 4755   |` cres 4756   "cima 4757   Fun wfun 5351   -->wf 5353   -1-1->wf1 5354   -onto->wfo 5355   -1-1-onto->wf1o 5356

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364

This theorem is referenced by:  f1opw2  6269  ssenen  7118  ballotfilemscr  13206  ballotfilemrinv0  13220  hmeoopn  15302  hmeocld  15303  hmeontr  15304