Theorem fvelimab 5485

Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)

Assertion

Ref	Expression
fvelimab	|- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. x e. B ( F ` x ) = C ) )

Distinct variable groups:   x, B   x, C   x, F

Allowed substitution hint:   A( x)

Proof of Theorem fvelimab

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2700	. . . 4 |- ( C e. ( F " B ) -> C e. _V )
2	1	anim2i 340	. . 3 |- ( ( ( F Fn A /\ B C_ A ) /\ C e. ( F " B ) ) -> ( ( F Fn A /\ B C_ A ) /\ C e. _V ) )
3		ssel2 3097	. . . . . . . 8 |- ( ( B C_ A /\ u e. B ) -> u e. A )
4		funfvex 5446	. . . . . . . . 9 |- ( ( Fun F /\ u e. dom F ) -> ( F ` u ) e. _V )
5	4	funfni 5231	. . . . . . . 8 |- ( ( F Fn A /\ u e. A ) -> ( F ` u ) e. _V )
6	3, 5	sylan2 284	. . . . . . 7 |- ( ( F Fn A /\ ( B C_ A /\ u e. B ) ) -> ( F ` u ) e. _V )
7	6	anassrs 398	. . . . . 6 |- ( ( ( F Fn A /\ B C_ A ) /\ u e. B ) -> ( F ` u ) e. _V )
8		eleq1 2203	. . . . . 6 |- ( ( F ` u ) = C -> ( ( F ` u ) e. _V <-> C e. _V ) )
9	7, 8	syl5ibcom 154	. . . . 5 |- ( ( ( F Fn A /\ B C_ A ) /\ u e. B ) -> ( ( F ` u ) = C -> C e. _V ) )
10	9	rexlimdva 2552	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( E. u e. B ( F ` u ) = C -> C e. _V ) )
11	10	imdistani 442	. . 3 |- ( ( ( F Fn A /\ B C_ A ) /\ E. u e. B ( F ` u ) = C ) -> ( ( F Fn A /\ B C_ A ) /\ C e. _V ) )
12		eleq1 2203	. . . . . . 7 |- ( v = C -> ( v e. ( F " B ) <-> C e. ( F " B ) ) )
13		eqeq2 2150	. . . . . . . 8 |- ( v = C -> ( ( F ` u ) = v <-> ( F ` u ) = C ) )
14	13	rexbidv 2439	. . . . . . 7 |- ( v = C -> ( E. u e. B ( F ` u ) = v <-> E. u e. B ( F ` u ) = C ) )
15	12, 14	bibi12d 234	. . . . . 6 |- ( v = C -> ( ( v e. ( F " B ) <-> E. u e. B ( F ` u ) = v ) <-> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) ) )
16	15	imbi2d 229	. . . . 5 |- ( v = C -> ( ( ( F Fn A /\ B C_ A ) -> ( v e. ( F " B ) <-> E. u e. B ( F ` u ) = v ) ) <-> ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) ) ) )
17		fnfun 5228	. . . . . . . 8 |- ( F Fn A -> Fun F )
18	17	adantr 274	. . . . . . 7 |- ( ( F Fn A /\ B C_ A ) -> Fun F )
19		fndm 5230	. . . . . . . . 9 |- ( F Fn A -> dom F = A )
20	19	sseq2d 3132	. . . . . . . 8 |- ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
21	20	biimpar 295	. . . . . . 7 |- ( ( F Fn A /\ B C_ A ) -> B C_ dom F )
22		dfimafn 5478	. . . . . . 7 |- ( ( Fun F /\ B C_ dom F ) -> ( F " B ) = { v | E. u e. B ( F ` u ) = v } )
23	18, 21, 22	syl2anc 409	. . . . . 6 |- ( ( F Fn A /\ B C_ A ) -> ( F " B ) = { v | E. u e. B ( F ` u ) = v } )
24	23	abeq2d 2253	. . . . 5 |- ( ( F Fn A /\ B C_ A ) -> ( v e. ( F " B ) <-> E. u e. B ( F ` u ) = v ) )
25	16, 24	vtoclg 2749	. . . 4 |- ( C e. _V -> ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) ) )
26	25	impcom 124	. . 3 |- ( ( ( F Fn A /\ B C_ A ) /\ C e. _V ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) )
27	2, 11, 26	pm5.21nd 902	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) )
28		fveq2 5429	. . . 4 |- ( u = x -> ( F ` u ) = ( F ` x ) )
29	28	eqeq1d 2149	. . 3 |- ( u = x -> ( ( F ` u ) = C <-> ( F ` x ) = C ) )
30	29	cbvrexv 2658	. 2 |- ( E. u e. B ( F ` u ) = C <-> E. x e. B ( F ` x ) = C )
31	27, 30	syl6bb 195	1 |- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. x e. B ( F ` x ) = C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   {cab 2126   E.wrex 2418   _Vcvv 2689   C_ wss 3076   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:  ssimaex  5490  foima2  5661  rexima  5664  ralima  5665  f1elima  5682  ovelimab  5929