Theorem fvimacnv 3790

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 3559 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 3717	. . . . . 6 |- (F` A) e. V
2	1	snss 2452	. . . . 5 |- ((F` A) e. B <-> {(F` A)} (_ B)
3		imass2 3417	. . . . 5 |- ({(F` A)} (_ B -> (`'F"{(F` A)}) (_ (`'F"B))
4	2, 3	sylbi 199	. . . 4 |- ((F` A) e. B -> (`'F"{(F` A)}) (_ (`'F"B))
5	4	sseld 2057	. . 3 |- ((F` A) e. B -> (A e. (`'F"{(F` A)}) -> A e. (`'F"B)))
6		funfvop 3788	. . . . 5 |- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)
7		opelcnvg 3285	. . . . . . 7 |- (((F` A) e. V /\ A e. dom F) -> (<.(F` A), A>. e. `'F <-> <.A, (F` A)>. e. F))
8	1, 7	mpan 693	. . . . . 6 |- (A e. dom F -> (<.(F` A), A>. e. `'F <-> <.A, (F` A)>. e. F))
9	8	adantl 388	. . . . 5 |- ((Fun F /\ A e. dom F) -> (<.(F` A), A>. e. `'F <-> <.A, (F` A)>. e. F))
10	6, 9	mpbird 196	. . . 4 |- ((Fun F /\ A e. dom F) -> <.(F` A), A>. e. `'F)
11		elimasng 3411	. . . . . 6 |- (((F` A) e. V /\ A e. dom F) -> (A e. (`'F"{(F` A)}) <-> <.(F` A), A>. e. `'F))
12	1, 11	mpan 693	. . . . 5 |- (A e. dom F -> (A e. (`'F"{(F` A)}) <-> <.(F` A), A>. e. `'F))
13	12	adantl 388	. . . 4 |- ((Fun F /\ A e. dom F) -> (A e. (`'F"{(F` A)}) <-> <.(F` A), A>. e. `'F))
14	10, 13	mpbird 196	. . 3 |- ((Fun F /\ A e. dom F) -> A e. (`'F"{(F` A)}))
15	5, 14	syl5com 52	. 2 |- ((Fun F /\ A e. dom F) -> ((F` A) e. B -> A e. (`'F"B)))
16		fvimacnvi 3789	. . . 4 |- ((Fun F /\ A e. (`'F"B)) -> (F` A) e. B)
17	16	ex 373	. . 3 |- (Fun F -> (A e. (`'F"B) -> (F` A) e. B))
18	17	adantr 389	. 2 |- ((Fun F /\ A e. dom F) -> (A e. (`'F"B) -> (F` A) e. B))
19	15, 18	impbid 514	1 |- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 955  Vcvv 1802   (_ wss 2037  {csn 2399  <.cop 2401  `'ccnv 3159  dom cdm 3160  "cima 3163  Fun wfun 3166  ` cfv 3172

This theorem is referenced by:  funimass3 3791  cnsscnp 7711  cncnplem4 7716

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

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