Theorem f1imacnv 4656

Description: Pre-image of an image.

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		df-f1 4011	. . . . . 6 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
2	1	simprbi 353	. . . . 5 |- (F:A-1-1->B -> Fun `'F)
3	2	adantr 425	. . . 4 |- ((F:A-1-1->B /\ C C_ A) -> Fun `'F)
4		funcnvres 4487	. . . 4 |- (Fun `'F -> `'(F |` C) = (`'F |` (F"C)))
5		imaeq1 4259	. . . 4 |- (`'(F |` C) = (`'F |` (F"C)) -> (`'(F |` C)"(F"C)) = ((`'F |` (F"C))"(F"C)))
6	3, 4, 5	3syl 24	. . 3 |- ((F:A-1-1->B /\ C C_ A) -> (`'(F |` C)"(F"C)) = ((`'F |` (F"C))"(F"C)))
7		f1ores 4654	. . . 4 |- ((F:A-1-1->B /\ C C_ A) -> (F |` C):C-1-1-onto->(F"C))
8		f1ocnv 4651	. . . 4 |- ((F |` C):C-1-1-onto->(F"C) -> `'(F |` C):(F"C)-1-1-onto->C)
9		f1of 4635	. . . . . . 7 |- (`'(F |` C):(F"C)-1-1-onto->C -> `'(F |` C):(F"C)-->C)
10		fdm 4567	. . . . . . 7 |- (`'(F |` C):(F"C)-->C -> dom `'(F |` C) = (F"C))
11		imaeq2 4260	. . . . . . 7 |- (dom `'(F |` C) = (F"C) -> (`'(F |` C)"dom `'(F |` C)) = (`'(F |` C)"(F"C)))
12	9, 10, 11	3syl 24	. . . . . 6 |- (`'(F |` C):(F"C)-1-1-onto->C -> (`'(F |` C)"dom `'(F |` C)) = (`'(F |` C)"(F"C)))
13		imadmrn 4277	. . . . . 6 |- (`'(F |` C)"dom `'(F |` C)) = ran `'(F |` C)
14	12, 13	syl5reqr 1943	. . . . 5 |- (`'(F |` C):(F"C)-1-1-onto->C -> (`'(F |` C)"(F"C)) = ran `'(F |` C))
15		f1ofo 4643	. . . . . 6 |- (`'(F |` C):(F"C)-1-1-onto->C -> `'(F |` C):(F"C)-onto->C)
16		forn 4620	. . . . . 6 |- (`'(F |` C):(F"C)-onto->C -> ran `'(F |` C) = C)
17	15, 16	syl 12	. . . . 5 |- (`'(F |` C):(F"C)-1-1-onto->C -> ran `'(F |` C) = C)
18	14, 17	eqtrd 1925	. . . 4 |- (`'(F |` C):(F"C)-1-1-onto->C -> (`'(F |` C)"(F"C)) = C)
19	7, 8, 18	3syl 24	. . 3 |- ((F:A-1-1->B /\ C C_ A) -> (`'(F |` C)"(F"C)) = C)
20	6, 19	eqtr3d 1927	. 2 |- ((F:A-1-1->B /\ C C_ A) -> ((`'F |` (F"C))"(F"C)) = C)
21		resima 4247	. 2 |- ((`'F |` (F"C))"(F"C)) = (`'F"(F"C))
22	20, 21	syl5eqr 1942	1 |- ((F:A-1-1->B /\ C C_ A) -> (`'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  -->wf 3994  -1-1->wf1 3995  -onto->wfo 3996  -1-1-onto->wf1o 3997

This theorem is referenced by:  ssenen 5598  f2imacnv 14355  oooeqim2 14356  compfipin0lem 15435  hmeoopn 15899  grpkerinj 16042

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-3an 860  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-f1 4011  df-fo 4012  df-f1o 4013

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