Theorem foimacnv 4657

Description: A reverse version of f1imacnv 4656. (Contributed by Jeffrey 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		fofun 4618	. . . . 5 |- (F:A-onto->B -> Fun F)
2	1	adantr 425	. . . 4 |- ((F:A-onto->B /\ C C_ B) -> Fun F)
3		funcnvres2 4489	. . . 4 |- (Fun F -> `'(`'F |` C) = (F |` (`'F"C)))
4		imaeq1 4259	. . . 4 |- (`'(`'F |` C) = (F |` (`'F"C)) -> (`'(`'F |` C)"(`'F"C)) = ((F |` (`'F"C))"(`'F"C)))
5	2, 3, 4	3syl 24	. . 3 |- ((F:A-onto->B /\ C C_ B) -> (`'(`'F |` C)"(`'F"C)) = ((F |` (`'F"C))"(`'F"C)))
6		df-fo 4012	. . . . 5 |- (`'(`'F |` C):(`'F"C)-onto->C <-> (`'(`'F |` C) Fn (`'F"C) /\ ran `'(`'F |` C) = C))
7		resss 4237	. . . . . . . . . . . 12 |- (`'F |` C) C_ `'F
8		cnvss 4134	. . . . . . . . . . . 12 |- ((`'F |` C) C_ `'F -> `'(`'F |` C) C_ `'`'F)
9	7, 8	ax-mp 7	. . . . . . . . . . 11 |- `'(`'F |` C) C_ `'`'F
10		cnvcnvss 4361	. . . . . . . . . . 11 |- `'`'F C_ F
11	9, 10	sstri 2626	. . . . . . . . . 10 |- `'(`'F |` C) C_ F
12		funss 4439	. . . . . . . . . 10 |- (`'(`'F |` C) C_ F -> (Fun F -> Fun `'(`'F |` C)))
13	11, 12	ax-mp 7	. . . . . . . . 9 |- (Fun F -> Fun `'(`'F |` C))
14	1, 13	syl 12	. . . . . . . 8 |- (F:A-onto->B -> Fun `'(`'F |` C))
15	14	adantr 425	. . . . . . 7 |- ((F:A-onto->B /\ C C_ B) -> Fun `'(`'F |` C))
16		df-ima 4007	. . . . . . . 8 |- (`'F"C) = ran (`'F |` C)
17		df-rn 4005	. . . . . . . 8 |- ran (`'F |` C) = dom `'(`'F |` C)
18	16, 17	eqtr2i 1909	. . . . . . 7 |- dom `'(`'F |` C) = (`'F"C)
19	15, 18	jctir 317	. . . . . 6 |- ((F:A-onto->B /\ C C_ B) -> (Fun `'(`'F |` C) /\ dom `'(`'F |` C) = (`'F"C)))
20		df-fn 4009	. . . . . 6 |- (`'(`'F |` C) Fn (`'F"C) <-> (Fun `'(`'F |` C) /\ dom `'(`'F |` C) = (`'F"C)))
21	19, 20	sylibr 217	. . . . 5 |- ((F:A-onto->B /\ C C_ B) -> `'(`'F |` C) Fn (`'F"C))
22		forn 4620	. . . . . . . . . 10 |- (F:A-onto->B -> ran F = B)
23	22	sseq2d 2645	. . . . . . . . 9 |- (F:A-onto->B -> (C C_ ran F <-> C C_ B))
24	23	biimpar 461	. . . . . . . 8 |- ((F:A-onto->B /\ C C_ B) -> C C_ ran F)
25		df-rn 4005	. . . . . . . 8 |- ran F = dom `' F
26	24, 25	syl6ss 2663	. . . . . . 7 |- ((F:A-onto->B /\ C C_ B) -> C C_ dom `' F)
27		ssdmres 4235	. . . . . . 7 |- (C C_ dom `' F <-> dom (`'F |` C) = C)
28	26, 27	sylib 215	. . . . . 6 |- ((F:A-onto->B /\ C C_ B) -> dom (`'F |` C) = C)
29		dfdm4 4151	. . . . . 6 |- dom (`'F |` C) = ran `'(`'F |` C)
30	28, 29	syl5eqr 1942	. . . . 5 |- ((F:A-onto->B /\ C C_ B) -> ran `'(`'F |` C) = C)
31	6, 21, 30	sylanbrc 527	. . . 4 |- ((F:A-onto->B /\ C C_ B) -> `'(`'F |` C):(`'F"C)-onto->C)
32		foima 4622	. . . 4 |- (`'(`'F |` C):(`'F"C)-onto->C -> (`'(`'F |` C)"(`'F"C)) = C)
33	31, 32	syl 12	. . 3 |- ((F:A-onto->B /\ C C_ B) -> (`'(`'F |` C)"(`'F"C)) = C)
34	5, 33	eqtr3d 1927	. 2 |- ((F:A-onto->B /\ C C_ B) -> ((F |` (`'F"C))"(`'F"C)) = C)
35		resima 4247	. 2 |- ((F |` (`'F"C))"(`'F"C)) = (F"(`'F"C))
36	34, 35	syl5eqr 1942	1 |- ((F:A-onto->B /\ C C_ B) -> (F"(`'F"C)) = C)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   C_ wss 2593  `'ccnv 3985  dom cdm 3986  ran crn 3987   |` cres 3988  "cima 3989  Fun wfun 3992   Fn wfn 3993  -onto->wfo 3996

This theorem is referenced by:  elfilmap3 10314  cncomp 10331  cnconn 15444

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012

Copyright terms: Public domain