Theorem foimacnv 5393

Description: A reverse version of f1imacnv 5392. (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 4860	. 2 |- ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) = ( F " ( `' F " C ) )
2		fofun 5354	. . . . . 6 |- ( F : A -onto-> B -> Fun F )
3	2	adantr 274	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> Fun F )
4		funcnvres2 5206	. . . . 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 4888	. . 3 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( `' ( `' F |` C ) " ( `' F " C ) ) = ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) )
7		resss 4851	. . . . . . . . . . 11 |- ( `' F |` C ) C_ `' F
8		cnvss 4720	. . . . . . . . . . 11 |- ( ( `' F |` C ) C_ `' F -> `' ( `' F |` C ) C_ `' `' F )
9	7, 8	ax-mp 5	. . . . . . . . . 10 |- `' ( `' F |` C ) C_ `' `' F
10		cnvcnvss 5001	. . . . . . . . . 10 |- `' `' F C_ F
11	9, 10	sstri 3111	. . . . . . . . 9 |- `' ( `' F |` C ) C_ F
12		funss 5150	. . . . . . . . 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 4560	. . . . . . . 8 |- ( `' F " C ) = ran ( `' F |` C )
16		df-rn 4558	. . . . . . . 8 |- ran ( `' F |` C ) = dom `' ( `' F |` C )
17	15, 16	eqtr2i 2162	. . . . . . 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 5134	. . . . . 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 4739	. . . . . 6 |- dom ( `' F |` C ) = ran `' ( `' F |` C )
22		forn 5356	. . . . . . . . . 10 |- ( F : A -onto-> B -> ran F = B )
23	22	sseq2d 3132	. . . . . . . . 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 4558	. . . . . . . 8 |- ran F = dom `' F
26	24, 25	sseqtrdi 3150	. . . . . . 7 |- ( ( F : A -onto-> B /\ C C_ B ) -> C C_ dom `' F )
27		ssdmres 4849	. . . . . . 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	syl5eqr 2187	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> ran `' ( `' F |` C ) = C )
30		df-fo 5137	. . . . 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 5358	. . . 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 2175	. 2 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) = C )
35	1, 34	syl5eqr 2187	1 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( F " ( `' F " C ) ) = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   C_ wss 3076   `'ccnv 4546   dom cdm 4547   ran crn 4548   |` cres 4549   "cima 4550   Fun wfun 5125   Fn wfn 5126   -onto->wfo 5129

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137

This theorem is referenced by:  f1opw2  5984  fopwdom  6738  fisumss  11193  hmeoimaf1o  12522