Theorem f1imacnv 5302

Description: Preimage of an image. (Contributed by set.mm contributors, 30-Sep-2004.)

Assertion

Ref	Expression
f1imacnv	|- ((F:A-1-1->B /\ C C_ A) -> (`'F"(F"C)) = C)

Proof of Theorem f1imacnv

Step	Hyp	Ref	Expression
1		resima 5006	. 2 |- ((`'F |` (F"C))"(F"C)) = (`'F"(F"C))
2		df-f1 4792	. . . . . 6 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
3	2	simprbi 450	. . . . 5 |- (F:A-1-1->B -> Fun `'F)
4	3	adantr 451	. . . 4 |- ((F:A-1-1->B /\ C C_ A) -> Fun `'F)
5		funcnvres 5165	. . . 4 |- (Fun `'F -> `'(F |` C) = (`'F |` (F"C)))
6		imaeq1 4937	. . . 4 |- (`'(F |` C) = (`'F |` (F"C)) -> (`'(F |` C)"(F"C)) = ((`'F |` (F"C))"(F"C)))
7	4, 5, 6	3syl 18	. . 3 |- ((F:A-1-1->B /\ C C_ A) -> (`'(F |` C)"(F"C)) = ((`'F |` (F"C))"(F"C)))
8		f1ores 5300	. . . 4 |- ((F:A-1-1->B /\ C C_ A) -> (F |` C):C-1-1-onto->(F"C))
9		f1ocnv 5299	. . . 4 |- ((F |` C):C-1-1-onto->(F"C) -> `'(F |` C):(F"C)-1-1-onto->C)
10		imadmrn 5008	. . . . . 6 |- (`'(F |` C)"dom `'(F |` C)) = ran `'(F |` C)
11		f1of 5287	. . . . . . 7 |- (`'(F |` C):(F"C)-1-1-onto->C -> `'(F |` C):(F"C)-->C)
12		fdm 5226	. . . . . . 7 |- (`'(F |` C):(F"C)-->C -> dom `'(F |` C) = (F"C))
13		imaeq2 4938	. . . . . . 7 |- (dom `'(F |` C) = (F"C) -> (`'(F |` C)"dom `'(F |` C)) = (`'(F |` C)"(F"C)))
14	11, 12, 13	3syl 18	. . . . . 6 |- (`'(F |` C):(F"C)-1-1-onto->C -> (`'(F |` C)"dom `'(F |` C)) = (`'(F |` C)"(F"C)))
15	10, 14	syl5reqr 2400	. . . . 5 |- (`'(F |` C):(F"C)-1-1-onto->C -> (`'(F |` C)"(F"C)) = ran `'(F |` C))
16		f1ofo 5293	. . . . . 6 |- (`'(F |` C):(F"C)-1-1-onto->C -> `'(F |` C):(F"C)-onto->C)
17		forn 5272	. . . . . 6 |- (`'(F |` C):(F"C)-onto->C -> ran `'(F |` C) = C)
18	16, 17	syl 15	. . . . 5 |- (`'(F |` C):(F"C)-1-1-onto->C -> ran `'(F |` C) = C)
19	15, 18	eqtrd 2385	. . . 4 |- (`'(F |` C):(F"C)-1-1-onto->C -> (`'(F |` C)"(F"C)) = C)
20	8, 9, 19	3syl 18	. . 3 |- ((F:A-1-1->B /\ C C_ A) -> (`'(F |` C)"(F"C)) = C)
21	7, 20	eqtr3d 2387	. 2 |- ((F:A-1-1->B /\ C C_ A) -> ((`'F |` (F"C))"(F"C)) = C)
22	1, 21	syl5eqr 2399	1 |- ((F:A-1-1->B /\ C C_ A) -> (`'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  -->wf 4777  -1-1->wf1 4778  -onto->wfo 4779  -1-1-onto->wf1o 4780

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-f1 4792  df-fo 4793  df-f1o 4794

This theorem is referenced by: (None)