Theorem fvimacnvi 5672

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 3762	. . 3 |- ( A e. ( `' F " B ) -> { A } C_ ( `' F " B ) )
2		funimass2 5332	. . 3 |- ( ( Fun F /\ { A } C_ ( `' F " B ) ) -> ( F " { A } ) C_ B )
3	1, 2	sylan2 286	. 2 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F " { A } ) C_ B )
4		cnvimass 5028	. . . . 5 |- ( `' F " B ) C_ dom F
5	4	sseli 3175	. . . 4 |- ( A e. ( `' F " B ) -> A e. dom F )
6		funfvex 5571	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
7		snssg 3752	. . . . 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 286	. . 3 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
10		funfn 5284	. . . . . 6 |- ( Fun F <-> F Fn dom F )
11		fnsnfv 5616	. . . . . 6 |- ( ( F Fn dom F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
12	10, 11	sylanb 284	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
13	5, 12	sylan2 286	. . . 4 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> { ( F ` A ) } = ( F " { A } ) )
14	13	sseq1d 3208	. . 3 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( { ( F ` A ) } C_ B <-> ( F " { A } ) C_ B ) )
15	9, 14	bitrd 188	. 2 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( ( F ` A ) e. B <-> ( F " { A } ) C_ B ) )
16	3, 15	mpbird 167	1 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   {csn 3618   `'ccnv 4658   dom cdm 4659   "cima 4662   Fun wfun 5248   Fn wfn 5249   ` cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  fvimacnv  5673  elpreima  5677  psrbaglesuppg  14158