Theorem fvimacnv 5695

Description: The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5352 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)

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		funfvop 5692	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
2		funfvex 5593	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
3		opelcnvg 4858	. . . . . 6 |- ( ( ( F ` A ) e. _V /\ A e. dom F ) -> ( <. ( F ` A ) , A >. e. `' F <-> <. A , ( F ` A ) >. e. F ) )
4	2, 3	sylancom 420	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( <. ( F ` A ) , A >. e. `' F <-> <. A , ( F ` A ) >. e. F ) )
5	1, 4	mpbird 167	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> <. ( F ` A ) , A >. e. `' F )
6		elimasng 5050	. . . . 5 |- ( ( ( F ` A ) e. _V /\ A e. dom F ) -> ( A e. ( `' F " { ( F ` A ) } ) <-> <. ( F ` A ) , A >. e. `' F ) )
7	2, 6	sylancom 420	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> ( A e. ( `' F " { ( F ` A ) } ) <-> <. ( F ` A ) , A >. e. `' F ) )
8	5, 7	mpbird 167	. . 3 |- ( ( Fun F /\ A e. dom F ) -> A e. ( `' F " { ( F ` A ) } ) )
9		snssg 3767	. . . . . . . 8 |- ( ( F ` A ) e. _V -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
10	2, 9	syl 14	. . . . . . 7 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
11		imass2 5058	. . . . . . 7 |- ( { ( F ` A ) } C_ B -> ( `' F " { ( F ` A ) } ) C_ ( `' F " B ) )
12	10, 11	biimtrdi 163	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B -> ( `' F " { ( F ` A ) } ) C_ ( `' F " B ) ) )
13	12	imp 124	. . . . 5 |- ( ( ( Fun F /\ A e. dom F ) /\ ( F ` A ) e. B ) -> ( `' F " { ( F ` A ) } ) C_ ( `' F " B ) )
14	13	sseld 3192	. . . 4 |- ( ( ( Fun F /\ A e. dom F ) /\ ( F ` A ) e. B ) -> ( A e. ( `' F " { ( F ` A ) } ) -> A e. ( `' F " B ) ) )
15	14	ex 115	. . 3 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B -> ( A e. ( `' F " { ( F ` A ) } ) -> A e. ( `' F " B ) ) ) )
16	8, 15	mpid 42	. 2 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B -> A e. ( `' F " B ) ) )
17		fvimacnvi 5694	. . . 4 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B )
18	17	ex 115	. . 3 |- ( Fun F -> ( A e. ( `' F " B ) -> ( F ` A ) e. B ) )
19	18	adantr 276	. 2 |- ( ( Fun F /\ A e. dom F ) -> ( A e. ( `' F " B ) -> ( F ` A ) e. B ) )
20	16, 19	impbid 129	1 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B <-> A e. ( `' F " B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2176   _Vcvv 2772   C_ wss 3166   {csn 3633   <.cop 3636   `'ccnv 4674   dom cdm 4675   "cima 4678   Fun wfun 5265   ` cfv 5271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279

This theorem is referenced by:  funimass3  5696  elpreima  5699  fisumss  11703  psrbaglesuppg  14434