Theorem fvimacnv 3805

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 3573 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 3732	. . . . . 6 |- (F` A) e. V
2	1	snss 2461	. . . . 5 |- ((F` A) e. B <-> {(F` A)} (_ B)
3		imass2 3433	. . . . 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 2067	. . 3 |- ((F` A) e. B -> (A e. (`'F"{(F` A)}) -> A e. (`'F"B)))
6		funfvop 3803	. . . . 5 |- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)
7		opelcnvg 3296	. . . . . . 7 |- (((F` A) e. V /\ A e. dom F) -> (<.(F` A), A>. e. `'F <-> <.A, (F` A)>. e. F))
8	1, 7	mpan 695	. . . . . 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 3427	. . . . . 6 |- (((F` A) e. V /\ A e. dom F) -> (A e. (`'F"{(F` A)}) <-> <.(F` A), A>. e. `'F))
12	1, 11	mpan 695	. . . . 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 3804	. . . 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 516	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 958  Vcvv 1811   (_ wss 2047  {csn 2409  <.cop 2411  `'ccnv 3169  dom cdm 3170  "cima 3173  Fun wfun 3176  ` cfv 3182

This theorem is referenced by:  funimass3 3806  cnsscnp 7772  cncnplem4 7777

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  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-uni 2504  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-fv 3198

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