Theorem fvimacnv 4778

Description: The argument of a function value belongs to the pre-image of any class containing the function value. (Contributed by Raph Levien, 20-Nov-2006. He remarks: "This proof is unsatisfying, because it seems to me that funimass2 4492 could probably be strengthened to a biconditional.")

Assertion

Ref	Expression
fvimacnv	|- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))

Proof of Theorem fvimacnv

Step	Hyp	Ref	Expression
1		fvex 4689	. . . . . 6 |- (F` A) e. _V
2	1	snss 3122	. . . . 5 |- ((F` A) e. B <-> {(F` A)} C_ B)
3		imass2 4299	. . . . 5 |- ({(F` A)} C_ B -> (`'F"{(F` A)}) C_ (`'F"B))
4	2, 3	sylbi 216	. . . 4 |- ((F` A) e. B -> (`'F"{(F` A)}) C_ (`'F"B))
5	4	sseld 2619	. . 3 |- ((F` A) e. B -> (A e. (`'F"{(F` A)}) -> A e. (`'F"B)))
6		funfvop 4776	. . . . 5 |- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)
7		opelcnvg 4140	. . . . . . 7 |- (((F` A) e. _V /\ A e. dom F) -> (<.(F` A), A>. e. `'F <-> <.A, (F` A)>. e. F))
8	1, 7	mpan 759	. . . . . 6 |- (A e. dom F -> (<.(F` A), A>. e. `'F <-> <.A, (F` A)>. e. F))
9	8	adantl 424	. . . . 5 |- ((Fun F /\ A e. dom F) -> (<.(F` A), A>. e. `'F <-> <.A, (F` A)>. e. F))
10	6, 9	mpbird 213	. . . 4 |- ((Fun F /\ A e. dom F) -> <.(F` A), A>. e. `'F)
11		elimasng 4291	. . . . . 6 |- (((F` A) e. _V /\ A e. dom F) -> (A e. (`'F"{(F` A)}) <-> <.(F` A), A>. e. `'F))
12	1, 11	mpan 759	. . . . 5 |- (A e. dom F -> (A e. (`'F"{(F` A)}) <-> <.(F` A), A>. e. `'F))
13	12	adantl 424	. . . 4 |- ((Fun F /\ A e. dom F) -> (A e. (`'F"{(F` A)}) <-> <.(F` A), A>. e. `'F))
14	10, 13	mpbird 213	. . 3 |- ((Fun F /\ A e. dom F) -> A e. (`'F"{(F` A)}))
15	5, 14	syl5com 63	. 2 |- ((Fun F /\ A e. dom F) -> ((F` A) e. B -> A e. (`'F"B)))
16		fvimacnvi 4777	. . . 4 |- ((Fun F /\ A e. (`'F"B)) -> (F` A) e. B)
17	16	ex 402	. . 3 |- (Fun F -> (A e. (`'F"B) -> (F` A) e. B))
18	17	adantr 425	. 2 |- ((Fun F /\ A e. dom F) -> (A e. (`'F"B) -> (F` A) e. B))
19	15, 18	impbid 574	1 |- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  _Vcvv 2292   C_ wss 2593  {csn 3044  <.cop 3046  `'ccnv 3985  dom cdm 3986  "cima 3989  Fun wfun 3992  ` cfv 3998

This theorem is referenced by:  funimass3 4779  cnsscnp 9049  cncnplem4 9054  elpreima 10161  inpreima2 14468  inpreima5 14469  rnelfmlem 15592  rnelfm 15593  fmfnfmlem2 15595  fmfnfmlem4 15597  fmfnfm 15598  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-13 1311  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  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-uni 3178  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-fv 4014

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