Theorem f1imacnv 3705

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 3195	. . . . . 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 389	. . . 4 |- ((F:A-1-1->B /\ C (_ A) -> Fun `'F)
4		funcnvres 3568	. . . 4 |- (Fun `'F -> `'(F |` C) = (`'F |` (F"C)))
5		imaeq1 3401	. . . 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 3703	. . . 4 |- ((F:A-1-1->B /\ C (_ A) -> (F |` C):C-1-1-onto->(F"C))
8		f1ocnv 3701	. . . 4 |- ((F |` C):C-1-1-onto->(F"C) -> `'(F |` C):(F"C)-1-1-onto->C)
9		f1of 3689	. . . . . . 7 |- (`'(F |` C):(F"C)-1-1-onto->C -> `'(F |` C):(F"C)-->C)
10		fdm 3631	. . . . . . 7 |- (`'(F |` C):(F"C)-->C -> dom `'(F |` C) = (F"C))
11		imaeq2 3402	. . . . . . 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 3414	. . . . . 6 |- (`'(F |` C)"dom `'(F |` C)) = ran `'(F |` C)
14	12, 13	syl5reqr 1522	. . . . 5 |- (`'(F |` C):(F"C)-1-1-onto->C -> (`'(F |` C)"(F"C)) = ran `'(F |` C))
15		f1ofo 3695	. . . . . 6 |- (`'(F |` C):(F"C)-1-1-onto->C -> `'(F |` C):(F"C)-onto->C)
16		forn 3674	. . . . . 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 1507	. . . 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 1509	. 2 |- ((F:A-1-1->B /\ C (_ A) -> ((`'F |` (F"C))"(F"C)) = C)
21		resima 3391	. 2 |- ((`'F |` (F"C))"(F"C)) = (`'F"(F"C))
22	20, 21	syl5eqr 1521	1 |- ((F:A-1-1->B /\ C (_ A) -> (`'F"(F"C)) = C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   (_ wss 2047  `'ccnv 3169  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173  Fun wfun 3176  -->wf 3178  -1-1->wf1 3179  -onto->wfo 3180  -1-1-onto->wf1o 3181

This theorem is referenced by:  ssenen 4504  f2imacnv 10475  oooeqim2 10476

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197

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