Theorem fvelimab 5470

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 2692	. . . 4 |- ( C e. ( F " B ) -> C e. _V )
2	1	anim2i 339	. . 3 |- ( ( ( F Fn A /\ B C_ A ) /\ C e. ( F " B ) ) -> ( ( F Fn A /\ B C_ A ) /\ C e. _V ) )
3		ssel2 3087	. . . . . . . 8 |- ( ( B C_ A /\ u e. B ) -> u e. A )
4		funfvex 5431	. . . . . . . . 9 |- ( ( Fun F /\ u e. dom F ) -> ( F ` u ) e. _V )
5	4	funfni 5218	. . . . . . . 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 397	. . . . . 6 |- ( ( ( F Fn A /\ B C_ A ) /\ u e. B ) -> ( F ` u ) e. _V )
8		eleq1 2200	. . . . . 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 2547	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( E. u e. B ( F ` u ) = C -> C e. _V ) )
11	10	imdistani 441	. . 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 2200	. . . . . . 7 |- ( v = C -> ( v e. ( F " B ) <-> C e. ( F " B ) ) )
13		eqeq2 2147	. . . . . . . 8 |- ( v = C -> ( ( F ` u ) = v <-> ( F ` u ) = C ) )
14	13	rexbidv 2436	. . . . . . 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 5215	. . . . . . . 8 |- ( F Fn A -> Fun F )
18	17	adantr 274	. . . . . . 7 |- ( ( F Fn A /\ B C_ A ) -> Fun F )
19		fndm 5217	. . . . . . . . 9 |- ( F Fn A -> dom F = A )
20	19	sseq2d 3122	. . . . . . . 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 5463	. . . . . . 7 |- ( ( Fun F /\ B C_ dom F ) -> ( F " B ) = { v | E. u e. B ( F ` u ) = v } )
23	18, 21, 22	syl2anc 408	. . . . . 6 |- ( ( F Fn A /\ B C_ A ) -> ( F " B ) = { v | E. u e. B ( F ` u ) = v } )
24	23	abeq2d 2250	. . . . 5 |- ( ( F Fn A /\ B C_ A ) -> ( v e. ( F " B ) <-> E. u e. B ( F ` u ) = v ) )
25	16, 24	vtoclg 2741	. . . 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 901	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) )
28		fveq2 5414	. . . 4 |- ( u = x -> ( F ` u ) = ( F ` x ) )
29	28	eqeq1d 2146	. . 3 |- ( u = x -> ( ( F ` u ) = C <-> ( F ` x ) = C ) )
30	29	cbvrexv 2653	. 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 1331   e. wcel 1480   {cab 2123   E.wrex 2415   _Vcvv 2681   C_ wss 3066   dom cdm 4534   "cima 4537   Fun wfun 5112   Fn wfn 5113   ` cfv 5118

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by:  ssimaex  5475  foima2  5646  rexima  5649  ralima  5650  f1elima  5667  ovelimab  5914