Theorem foimacnv 5657

Description: A reverse version of f1imacnv 5656. (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 5096	. 2 |- ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) = ( F " ( `' F " C ) )
2		fofun 5616	. . . . . 6 |- ( F : A -onto-> B -> Fun F )
3	2	adantr 276	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> Fun F )
4		funcnvres2 5456	. . . . 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 5125	. . 3 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( `' ( `' F |` C ) " ( `' F " C ) ) = ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) )
7		resss 5087	. . . . . . . . . . 11 |- ( `' F |` C ) C_ `' F
8		cnvss 4953	. . . . . . . . . . 11 |- ( ( `' F |` C ) C_ `' F -> `' ( `' F |` C ) C_ `' `' F )
9	7, 8	ax-mp 5	. . . . . . . . . 10 |- `' ( `' F |` C ) C_ `' `' F
10		cnvcnvss 5242	. . . . . . . . . 10 |- `' `' F C_ F
11	9, 10	sstri 3257	. . . . . . . . 9 |- `' ( `' F |` C ) C_ F
12		funss 5396	. . . . . . . . 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 276	. . . . . . 7 |- ( ( F : A -onto-> B /\ C C_ B ) -> Fun `' ( `' F |` C ) )
15		df-ima 4787	. . . . . . . 8 |- ( `' F " C ) = ran ( `' F |` C )
16		df-rn 4785	. . . . . . . 8 |- ran ( `' F |` C ) = dom `' ( `' F |` C )
17	15, 16	eqtr2i 2260	. . . . . . 7 |- dom `' ( `' F |` C ) = ( `' F " C )
18	14, 17	jctir 313	. . . . . 6 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( Fun `' ( `' F |` C ) /\ dom `' ( `' F |` C ) = ( `' F " C ) ) )
19		df-fn 5380	. . . . . 6 |- ( `' ( `' F |` C ) Fn ( `' F " C ) <-> ( Fun `' ( `' F |` C ) /\ dom `' ( `' F |` C ) = ( `' F " C ) ) )
20	18, 19	sylibr 134	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> `' ( `' F |` C ) Fn ( `' F " C ) )
21		dfdm4 4973	. . . . . 6 |- dom ( `' F |` C ) = ran `' ( `' F |` C )
22		forn 5618	. . . . . . . . . 10 |- ( F : A -onto-> B -> ran F = B )
23	22	sseq2d 3278	. . . . . . . . 9 |- ( F : A -onto-> B -> ( C C_ ran F <-> C C_ B ) )
24	23	biimpar 297	. . . . . . . 8 |- ( ( F : A -onto-> B /\ C C_ B ) -> C C_ ran F )
25		df-rn 4785	. . . . . . . 8 |- ran F = dom `' F
26	24, 25	sseqtrdi 3296	. . . . . . 7 |- ( ( F : A -onto-> B /\ C C_ B ) -> C C_ dom `' F )
27		ssdmres 5085	. . . . . . 7 |- ( C C_ dom `' F <-> dom ( `' F |` C ) = C )
28	26, 27	sylib 122	. . . . . 6 |- ( ( F : A -onto-> B /\ C C_ B ) -> dom ( `' F |` C ) = C )
29	21, 28	eqtr3id 2285	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> ran `' ( `' F |` C ) = C )
30		df-fo 5383	. . . . 5 |- ( `' ( `' F |` C ) : ( `' F " C ) -onto-> C <-> ( `' ( `' F |` C ) Fn ( `' F " C ) /\ ran `' ( `' F |` C ) = C ) )
31	20, 29, 30	sylanbrc 421	. . . 4 |- ( ( F : A -onto-> B /\ C C_ B ) -> `' ( `' F |` C ) : ( `' F " C ) -onto-> C )
32		foima 5620	. . . 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 2273	. 2 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) = C )
35	1, 34	eqtr3id 2285	1 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( F " ( `' F " C ) ) = C )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   C_ wss 3220   `'ccnv 4773   dom cdm 4774   ran crn 4775   |` cres 4776   "cima 4777   Fun wfun 5371   Fn wfn 5372   -onto->wfo 5375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383

This theorem is used by:  f1opw2  6296  imacosuppfn  6508  fopwdom  7136  fisumss  12159  fprodssdc  12357  hmeoimaf1o  15415