Theorem fvimacnvi 5542

Description: A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)

Assertion

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

Proof of Theorem fvimacnvi

Step	Hyp	Ref	Expression
1		snssi 3672	. . 3 |- ( A e. ( `' F " B ) -> { A } C_ ( `' F " B ) )
2		funimass2 5209	. . 3 |- ( ( Fun F /\ { A } C_ ( `' F " B ) ) -> ( F " { A } ) C_ B )
3	1, 2	sylan2 284	. 2 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F " { A } ) C_ B )
4		cnvimass 4910	. . . . 5 |- ( `' F " B ) C_ dom F
5	4	sseli 3098	. . . 4 |- ( A e. ( `' F " B ) -> A e. dom F )
6		funfvex 5446	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
7		snssg 3664	. . . . 5 |- ( ( F ` A ) e. _V -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
8	6, 7	syl 14	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
9	5, 8	sylan2 284	. . 3 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
10		funfn 5161	. . . . . 6 |- ( Fun F <-> F Fn dom F )
11		fnsnfv 5488	. . . . . 6 |- ( ( F Fn dom F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
12	10, 11	sylanb 282	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
13	5, 12	sylan2 284	. . . 4 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> { ( F ` A ) } = ( F " { A } ) )
14	13	sseq1d 3131	. . 3 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( { ( F ` A ) } C_ B <-> ( F " { A } ) C_ B ) )
15	9, 14	bitrd 187	. 2 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( ( F ` A ) e. B <-> ( F " { A } ) C_ B ) )
16	3, 15	mpbird 166	1 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   _Vcvv 2689   C_ wss 3076   {csn 3532   `'ccnv 4546   dom cdm 4547   "cima 4550   Fun wfun 5125   Fn wfn 5126   ` cfv 5131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139

This theorem is referenced by:  fvimacnv  5543  elpreima  5547