Theorem foimacnv 5164

Description: A reverse version of f1imacnv 5163. (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 4661	. 2 |- ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) = ( F " ( `' F " C ) )
2		fofun 5127	. . . . . 6 |- ( F : A -onto-> B -> Fun F )
3	2	adantr 270	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> Fun F )
4		funcnvres2 4994	. . . . 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 4687	. . 3 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( `' ( `' F |` C ) " ( `' F " C ) ) = ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) )
7		resss 4653	. . . . . . . . . . 11 |- ( `' F |` C ) C_ `' F
8		cnvss 4526	. . . . . . . . . . 11 |- ( ( `' F |` C ) C_ `' F -> `' ( `' F |` C ) C_ `' `' F )
9	7, 8	ax-mp 7	. . . . . . . . . 10 |- `' ( `' F |` C ) C_ `' `' F
10		cnvcnvss 4795	. . . . . . . . . 10 |- `' `' F C_ F
11	9, 10	sstri 3008	. . . . . . . . 9 |- `' ( `' F |` C ) C_ F
12		funss 4940	. . . . . . . . 9 |- ( `' ( `' F |` C ) C_ F -> ( Fun F -> Fun `' ( `' F |` C ) ) )
13	11, 2, 12	mpsyl 64	. . . . . . . 8 |- ( F : A -onto-> B -> Fun `' ( `' F |` C ) )
14	13	adantr 270	. . . . . . 7 |- ( ( F : A -onto-> B /\ C C_ B ) -> Fun `' ( `' F |` C ) )
15		df-ima 4376	. . . . . . . 8 |- ( `' F " C ) = ran ( `' F |` C )
16		df-rn 4374	. . . . . . . 8 |- ran ( `' F |` C ) = dom `' ( `' F |` C )
17	15, 16	eqtr2i 2102	. . . . . . 7 |- dom `' ( `' F |` C ) = ( `' F " C )
18	14, 17	jctir 306	. . . . . 6 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( Fun `' ( `' F |` C ) /\ dom `' ( `' F |` C ) = ( `' F " C ) ) )
19		df-fn 4925	. . . . . 6 |- ( `' ( `' F |` C ) Fn ( `' F " C ) <-> ( Fun `' ( `' F |` C ) /\ dom `' ( `' F |` C ) = ( `' F " C ) ) )
20	18, 19	sylibr 132	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> `' ( `' F |` C ) Fn ( `' F " C ) )
21		dfdm4 4545	. . . . . 6 |- dom ( `' F |` C ) = ran `' ( `' F |` C )
22		forn 5129	. . . . . . . . . 10 |- ( F : A -onto-> B -> ran F = B )
23	22	sseq2d 3027	. . . . . . . . 9 |- ( F : A -onto-> B -> ( C C_ ran F <-> C C_ B ) )
24	23	biimpar 291	. . . . . . . 8 |- ( ( F : A -onto-> B /\ C C_ B ) -> C C_ ran F )
25		df-rn 4374	. . . . . . . 8 |- ran F = dom `' F
26	24, 25	syl6sseq 3045	. . . . . . 7 |- ( ( F : A -onto-> B /\ C C_ B ) -> C C_ dom `' F )
27		ssdmres 4651	. . . . . . 7 |- ( C C_ dom `' F <-> dom ( `' F |` C ) = C )
28	26, 27	sylib 120	. . . . . 6 |- ( ( F : A -onto-> B /\ C C_ B ) -> dom ( `' F |` C ) = C )
29	21, 28	syl5eqr 2127	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> ran `' ( `' F |` C ) = C )
30		df-fo 4928	. . . . 5 |- ( `' ( `' F |` C ) : ( `' F " C ) -onto-> C <-> ( `' ( `' F |` C ) Fn ( `' F " C ) /\ ran `' ( `' F |` C ) = C ) )
31	20, 29, 30	sylanbrc 408	. . . 4 |- ( ( F : A -onto-> B /\ C C_ B ) -> `' ( `' F |` C ) : ( `' F " C ) -onto-> C )
32		foima 5131	. . . 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 2115	. 2 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) = C )
35	1, 34	syl5eqr 2127	1 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( F " ( `' F " C ) ) = C )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   C_ wss 2973   `'ccnv 4362   dom cdm 4363   ran crn 4364   |` cres 4365   "cima 4366   Fun wfun 4916   Fn wfn 4917   -onto->wfo 4920

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928

This theorem is referenced by:  f1opw2  5726  fopwdom  6333