Theorem f1imacnv 3711

Description: Pre-image of an image.

Assertion

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

Proof of Theorem f1imacnv

Step	Hyp	Ref	Expression
1		df-f1 3201	. . . . . 6 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
2	1	pm3.27bi 326	. . . . 5 |- (F:A-1-1->B -> Fun `'F)
3	2	adantr 391	. . . 4 |- ((F:A-1-1->B /\ C (_ A) -> Fun `'F)
4		funcnvres 3574	. . . 4 |- (Fun `'F -> `'(F |` C) = (`'F |` (F"C)))
5		imaeq1 3407	. . . 4 |- (`'(F |` C) = (`'F |` (F"C)) -> (`'(F |` C)"(F"C)) = ((`'F |` (F"C))"(F"C)))
6	3, 4, 5	3syl 20	. . 3 |- ((F:A-1-1->B /\ C (_ A) -> (`'(F |` C)"(F"C)) = ((`'F |` (F"C))"(F"C)))
7		f1ores 3709	. . . 4 |- ((F:A-1-1->B /\ C (_ A) -> (F |` C):C-1-1-onto->(F"C))
8		f1ocnv 3707	. . . 4 |- ((F |` C):C-1-1-onto->(F"C) -> `'(F |` C):(F"C)-1-1-onto->C)
9		f1of 3695	. . . . . . 7 |- (`'(F |` C):(F"C)-1-1-onto->C -> `'(F |` C):(F"C)-->C)
10		fdm 3637	. . . . . . 7 |- (`'(F |` C):(F"C)-->C -> dom `'(F |` C) = (F"C))
11		imaeq2 3408	. . . . . . 7 |- (dom `'(F |` C) = (F"C) -> (`'(F |` C)"dom `'(F |` C)) = (`'(F |` C)"(F"C)))
12	9, 10, 11	3syl 20	. . . . . 6 |- (`'(F |` C):(F"C)-1-1-onto->C -> (`'(F |` C)"dom `'(F |` C)) = (`'(F |` C)"(F"C)))
13		imadmrn 3420	. . . . . 6 |- (`'(F |` C)"dom `'(F |` C)) = ran `'(F |` C)
14	12, 13	syl5reqr 1525	. . . . 5 |- (`'(F |` C):(F"C)-1-1-onto->C -> (`'(F |` C)"(F"C)) = ran `'(F |` C))
15		f1ofo 3701	. . . . . 6 |- (`'(F |` C):(F"C)-1-1-onto->C -> `'(F |` C):(F"C)-onto->C)
16		forn 3680	. . . . . 6 |- (`'(F |` C):(F"C)-onto->C -> ran `'(F |` C) = C)
17	15, 16	syl 10	. . . . 5 |- (`'(F |` C):(F"C)-1-1-onto->C -> ran `'(F |` C) = C)
18	14, 17	eqtrd 1510	. . . 4 |- (`'(F |` C):(F"C)-1-1-onto->C -> (`'(F |` C)"(F"C)) = C)
19	7, 8, 18	3syl 20	. . 3 |- ((F:A-1-1->B /\ C (_ A) -> (`'(F |` C)"(F"C)) = C)
20	6, 19	eqtr3d 1512	. 2 |- ((F:A-1-1->B /\ C (_ A) -> ((`'F |` (F"C))"(F"C)) = C)
21		resima 3397	. 2 |- ((`'F |` (F"C))"(F"C)) = (`'F"(F"C))
22	20, 21	syl5eqr 1524	1 |- ((F:A-1-1->B /\ C (_ A) -> (`'F"(F"C)) = C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   (_ wss 2050  `'ccnv 3175  dom cdm 3176  ran crn 3177   |` cres 3178  "cima 3179  Fun wfun 3182  -->wf 3184  -1-1->wf1 3185  -onto->wfo 3186  -1-1-onto->wf1o 3187

This theorem is referenced by:  ssenen 4510  f2imacnv 10464  oooeqim2 10465

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203

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