Theorem foimacnv 5303

Description: A reverse version of f1imacnv 5302. (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		resima 5006	. 2 |- ((F |` (`'F"C))"(`'F"C)) = (F"(`'F"C))
2		fofun 5270	. . . . . 6 |- (F:A-onto->B -> Fun F)
3	2	adantr 451	. . . . 5 |- ((F:A-onto->B /\ C C_ B) -> Fun F)
4		funcnvres2 5167	. . . . 5 |- (Fun F -> `'(`'F |` C) = (F |` (`'F"C)))
5	3, 4	syl 15	. . . 4 |- ((F:A-onto->B /\ C C_ B) -> `'(`'F |` C) = (F |` (`'F"C)))
6	5	imaeq1d 4941	. . 3 |- ((F:A-onto->B /\ C C_ B) -> (`'(`'F |` C)"(`'F"C)) = ((F |` (`'F"C))"(`'F"C)))
7		resss 4988	. . . . . . . . . . 11 |- (`'F |` C) C_ `'F
8		cnvss 4885	. . . . . . . . . . 11 |- ((`'F |` C) C_ `'F -> `'(`'F |` C) C_ `'`'F)
9	7, 8	ax-mp 5	. . . . . . . . . 10 |- `'(`'F |` C) C_ `'`'F
10		cnvcnv 5062	. . . . . . . . . 10 |- `'`'F = F
11	9, 10	sseqtri 3303	. . . . . . . . 9 |- `'(`'F |` C) C_ F
12		funss 5126	. . . . . . . . 9 |- (`'(`'F |` C) C_ F -> (Fun F -> Fun `'(`'F |` C)))
13	11, 2, 12	mpsyl 59	. . . . . . . 8 |- (F:A-onto->B -> Fun `'(`'F |` C))
14	13	adantr 451	. . . . . . 7 |- ((F:A-onto->B /\ C C_ B) -> Fun `'(`'F |` C))
15		dfima3 4951	. . . . . . . 8 |- (`'F"C) = ran (`'F |` C)
16		dfrn4 4904	. . . . . . . 8 |- ran (`'F |` C) = dom `'(`'F |` C)
17	15, 16	eqtr2i 2374	. . . . . . 7 |- dom `'(`'F |` C) = (`'F"C)
18	14, 17	jctir 524	. . . . . 6 |- ((F:A-onto->B /\ C C_ B) -> (Fun `'(`'F |` C) /\ dom `'(`'F |` C) = (`'F"C)))
19		df-fn 4790	. . . . . 6 |- (`'(`'F |` C) Fn (`'F"C) <-> (Fun `'(`'F |` C) /\ dom `'(`'F |` C) = (`'F"C)))
20	18, 19	sylibr 203	. . . . 5 |- ((F:A-onto->B /\ C C_ B) -> `'(`'F |` C) Fn (`'F"C))
21		df-dm 4787	. . . . . 6 |- dom (`'F |` C) = ran `'(`'F |` C)
22		forn 5272	. . . . . . . . . 10 |- (F:A-onto->B -> ran F = B)
23	22	sseq2d 3299	. . . . . . . . 9 |- (F:A-onto->B -> (C C_ ran F <-> C C_ B))
24	23	biimpar 471	. . . . . . . 8 |- ((F:A-onto->B /\ C C_ B) -> C C_ ran F)
25		dfrn4 4904	. . . . . . . 8 |- ran F = dom `'F
26	24, 25	syl6sseq 3317	. . . . . . 7 |- ((F:A-onto->B /\ C C_ B) -> C C_ dom `'F)
27		ssdmres 4987	. . . . . . 7 |- (C C_ dom `'F <-> dom (`'F |` C) = C)
28	26, 27	sylib 188	. . . . . 6 |- ((F:A-onto->B /\ C C_ B) -> dom (`'F |` C) = C)
29	21, 28	syl5eqr 2399	. . . . 5 |- ((F:A-onto->B /\ C C_ B) -> ran `'(`'F |` C) = C)
30		df-fo 4793	. . . . 5 |- (`'(`'F |` C):(`'F"C)-onto->C <-> (`'(`'F |` C) Fn (`'F"C) /\ ran `'(`'F |` C) = C))
31	20, 29, 30	sylanbrc 645	. . . 4 |- ((F:A-onto->B /\ C C_ B) -> `'(`'F |` C):(`'F"C)-onto->C)
32		foima 5274	. . . 4 |- (`'(`'F |` C):(`'F"C)-onto->C -> (`'(`'F |` C)"(`'F"C)) = C)
33	31, 32	syl 15	. . 3 |- ((F:A-onto->B /\ C C_ B) -> (`'(`'F |` C)"(`'F"C)) = C)
34	6, 33	eqtr3d 2387	. 2 |- ((F:A-onto->B /\ C C_ B) -> ((F |` (`'F"C))"(`'F"C)) = C)
35	1, 34	syl5eqr 2399	1 |- ((F:A-onto->B /\ C C_ B) -> (F"(`'F"C)) = C)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   C_ wss 3257  "cima 4722  `'ccnv 4771  dom cdm 4772  ran crn 4773   |` cres 4774  Fun wfun 4775   Fn wfn 4776  -onto->wfo 4779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793

This theorem is referenced by: (None)