Theorem foimacnv 5450

Description: A reverse version of f1imacnv 5449. (Contributed by Jeff Hankins, 16-Jul-2009.)

Assertion

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

Proof of Theorem foimacnv

Step	Hyp	Ref	Expression
1		resima 4917	. 2 |- ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) = ( F " ( `' F " C ) )
2		fofun 5411	. . . . . 6 |- ( F : A -onto-> B -> Fun F )
3	2	adantr 274	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> Fun F )
4		funcnvres2 5263	. . . . 5 |- ( Fun F -> `' ( `' F |` C ) = ( F |` ( `' F " C ) ) )
5	3, 4	syl 14	. . . 4 |- ( ( F : A -onto-> B /\ C C_ B ) -> `' ( `' F |` C ) = ( F |` ( `' F " C ) ) )
6	5	imaeq1d 4945	. . 3 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( `' ( `' F |` C ) " ( `' F " C ) ) = ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) )
7		resss 4908	. . . . . . . . . . 11 |- ( `' F |` C ) C_ `' F
8		cnvss 4777	. . . . . . . . . . 11 |- ( ( `' F |` C ) C_ `' F -> `' ( `' F |` C ) C_ `' `' F )
9	7, 8	ax-mp 5	. . . . . . . . . 10 |- `' ( `' F |` C ) C_ `' `' F
10		cnvcnvss 5058	. . . . . . . . . 10 |- `' `' F C_ F
11	9, 10	sstri 3151	. . . . . . . . 9 |- `' ( `' F |` C ) C_ F
12		funss 5207	. . . . . . . . 9 |- ( `' ( `' F |` C ) C_ F -> ( Fun F -> Fun `' ( `' F |` C ) ) )
13	11, 2, 12	mpsyl 65	. . . . . . . 8 |- ( F : A -onto-> B -> Fun `' ( `' F |` C ) )
14	13	adantr 274	. . . . . . 7 |- ( ( F : A -onto-> B /\ C C_ B ) -> Fun `' ( `' F |` C ) )
15		df-ima 4617	. . . . . . . 8 |- ( `' F " C ) = ran ( `' F |` C )
16		df-rn 4615	. . . . . . . 8 |- ran ( `' F |` C ) = dom `' ( `' F |` C )
17	15, 16	eqtr2i 2187	. . . . . . 7 |- dom `' ( `' F |` C ) = ( `' F " C )
18	14, 17	jctir 311	. . . . . 6 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( Fun `' ( `' F |` C ) /\ dom `' ( `' F |` C ) = ( `' F " C ) ) )
19		df-fn 5191	. . . . . 6 |- ( `' ( `' F |` C ) Fn ( `' F " C ) <-> ( Fun `' ( `' F |` C ) /\ dom `' ( `' F |` C ) = ( `' F " C ) ) )
20	18, 19	sylibr 133	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> `' ( `' F |` C ) Fn ( `' F " C ) )
21		dfdm4 4796	. . . . . 6 |- dom ( `' F |` C ) = ran `' ( `' F |` C )
22		forn 5413	. . . . . . . . . 10 |- ( F : A -onto-> B -> ran F = B )
23	22	sseq2d 3172	. . . . . . . . 9 |- ( F : A -onto-> B -> ( C C_ ran F <-> C C_ B ) )
24	23	biimpar 295	. . . . . . . 8 |- ( ( F : A -onto-> B /\ C C_ B ) -> C C_ ran F )
25		df-rn 4615	. . . . . . . 8 |- ran F = dom `' F
26	24, 25	sseqtrdi 3190	. . . . . . 7 |- ( ( F : A -onto-> B /\ C C_ B ) -> C C_ dom `' F )
27		ssdmres 4906	. . . . . . 7 |- ( C C_ dom `' F <-> dom ( `' F |` C ) = C )
28	26, 27	sylib 121	. . . . . 6 |- ( ( F : A -onto-> B /\ C C_ B ) -> dom ( `' F |` C ) = C )
29	21, 28	eqtr3id 2213	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> ran `' ( `' F |` C ) = C )
30		df-fo 5194	. . . . 5 |- ( `' ( `' F |` C ) : ( `' F " C ) -onto-> C <-> ( `' ( `' F |` C ) Fn ( `' F " C ) /\ ran `' ( `' F |` C ) = C ) )
31	20, 29, 30	sylanbrc 414	. . . 4 |- ( ( F : A -onto-> B /\ C C_ B ) -> `' ( `' F |` C ) : ( `' F " C ) -onto-> C )
32		foima 5415	. . . 4 |- ( `' ( `' F |` C ) : ( `' F " C ) -onto-> C -> ( `' ( `' F |` C ) " ( `' F " C ) ) = C )
33	31, 32	syl 14	. . 3 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( `' ( `' F |` C ) " ( `' F " C ) ) = C )
34	6, 33	eqtr3d 2200	. 2 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) = C )
35	1, 34	eqtr3id 2213	1 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( F " ( `' F " C ) ) = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   C_ wss 3116   `'ccnv 4603   dom cdm 4604   ran crn 4605   |` cres 4606   "cima 4607   Fun wfun 5182   Fn wfn 5183   -onto->wfo 5186

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194

This theorem is referenced by:  f1opw2  6044  fopwdom  6802  fisumss  11333  fprodssdc  11531  hmeoimaf1o  12954