Theorem foimacnv 5634

Description: A reverse version of f1imacnv 5633. (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 5073	. 2 |- ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) = ( F " ( `' F " C ) )
2		fofun 5593	. . . . . 6 |- ( F : A -onto-> B -> Fun F )
3	2	adantr 276	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> Fun F )
4		funcnvres2 5433	. . . . 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 5102	. . 3 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( `' ( `' F |` C ) " ( `' F " C ) ) = ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) )
7		resss 5064	. . . . . . . . . . 11 |- ( `' F |` C ) C_ `' F
8		cnvss 4930	. . . . . . . . . . 11 |- ( ( `' F |` C ) C_ `' F -> `' ( `' F |` C ) C_ `' `' F )
9	7, 8	ax-mp 5	. . . . . . . . . 10 |- `' ( `' F |` C ) C_ `' `' F
10		cnvcnvss 5219	. . . . . . . . . 10 |- `' `' F C_ F
11	9, 10	sstri 3249	. . . . . . . . 9 |- `' ( `' F |` C ) C_ F
12		funss 5373	. . . . . . . . 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 4764	. . . . . . . 8 |- ( `' F " C ) = ran ( `' F |` C )
16		df-rn 4762	. . . . . . . 8 |- ran ( `' F |` C ) = dom `' ( `' F |` C )
17	15, 16	eqtr2i 2256	. . . . . . 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 5357	. . . . . 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 4950	. . . . . 6 |- dom ( `' F |` C ) = ran `' ( `' F |` C )
22		forn 5595	. . . . . . . . . 10 |- ( F : A -onto-> B -> ran F = B )
23	22	sseq2d 3270	. . . . . . . . 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 4762	. . . . . . . 8 |- ran F = dom `' F
26	24, 25	sseqtrdi 3288	. . . . . . 7 |- ( ( F : A -onto-> B /\ C C_ B ) -> C C_ dom `' F )
27		ssdmres 5062	. . . . . . 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 2281	. . . . 5 |- ( ( F : A -onto-> B /\ C C_ B ) -> ran `' ( `' F |` C ) = C )
30		df-fo 5360	. . . . 5 |- ( `' ( `' F |` C ) : ( `' F " C ) -onto-> C <-> ( `' ( `' F |` C ) Fn ( `' F " C ) /\ ran `' ( `' F |` C ) = C ) )
31	20, 29, 30	sylanbrc 417	. . . 4 |- ( ( F : A -onto-> B /\ C C_ B ) -> `' ( `' F |` C ) : ( `' F " C ) -onto-> C )
32		foima 5597	. . . 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 2269	. 2 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( ( F |` ( `' F " C ) ) " ( `' F " C ) ) = C )
35	1, 34	eqtr3id 2281	1 |- ( ( F : A -onto-> B /\ C C_ B ) -> ( F " ( `' F " C ) ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   C_ wss 3213   `'ccnv 4750   dom cdm 4751   ran crn 4752   |` cres 4753   "cima 4754   Fun wfun 5348   Fn wfn 5349   -onto->wfo 5352

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 4230  ax-pow 4289  ax-pr 4324

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 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-fun 5356  df-fn 5357  df-f 5358  df-fo 5360

This theorem is referenced by:  f1opw2  6263  imacosuppfn  6470  fopwdom  7091  fisumss  12082  fprodssdc  12280  hmeoimaf1o  15196