Theorem fvelimab 5250

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 2610	. . . 4 |- ( C e. ( F " B ) -> C e. _V )
2	1	anim2i 334	. . 3 |- ( ( ( F Fn A /\ B C_ A ) /\ C e. ( F " B ) ) -> ( ( F Fn A /\ B C_ A ) /\ C e. _V ) )
3		ssel2 2994	. . . . . . . 8 |- ( ( B C_ A /\ u e. B ) -> u e. A )
4		funfvex 5212	. . . . . . . . 9 |- ( ( Fun F /\ u e. dom F ) -> ( F ` u ) e. _V )
5	4	funfni 5019	. . . . . . . 8 |- ( ( F Fn A /\ u e. A ) -> ( F ` u ) e. _V )
6	3, 5	sylan2 280	. . . . . . 7 |- ( ( F Fn A /\ ( B C_ A /\ u e. B ) ) -> ( F ` u ) e. _V )
7	6	anassrs 392	. . . . . 6 |- ( ( ( F Fn A /\ B C_ A ) /\ u e. B ) -> ( F ` u ) e. _V )
8		eleq1 2141	. . . . . 6 |- ( ( F ` u ) = C -> ( ( F ` u ) e. _V <-> C e. _V ) )
9	7, 8	syl5ibcom 153	. . . . 5 |- ( ( ( F Fn A /\ B C_ A ) /\ u e. B ) -> ( ( F ` u ) = C -> C e. _V ) )
10	9	rexlimdva 2477	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( E. u e. B ( F ` u ) = C -> C e. _V ) )
11	10	imdistani 433	. . 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 2141	. . . . . . 7 |- ( v = C -> ( v e. ( F " B ) <-> C e. ( F " B ) ) )
13		eqeq2 2090	. . . . . . . 8 |- ( v = C -> ( ( F ` u ) = v <-> ( F ` u ) = C ) )
14	13	rexbidv 2369	. . . . . . 7 |- ( v = C -> ( E. u e. B ( F ` u ) = v <-> E. u e. B ( F ` u ) = C ) )
15	12, 14	bibi12d 233	. . . . . 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 228	. . . . 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 5016	. . . . . . . 8 |- ( F Fn A -> Fun F )
18	17	adantr 270	. . . . . . 7 |- ( ( F Fn A /\ B C_ A ) -> Fun F )
19		fndm 5018	. . . . . . . . 9 |- ( F Fn A -> dom F = A )
20	19	sseq2d 3027	. . . . . . . 8 |- ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
21	20	biimpar 291	. . . . . . 7 |- ( ( F Fn A /\ B C_ A ) -> B C_ dom F )
22		dfimafn 5243	. . . . . . 7 |- ( ( Fun F /\ B C_ dom F ) -> ( F " B ) = { v | E. u e. B ( F ` u ) = v } )
23	18, 21, 22	syl2anc 403	. . . . . 6 |- ( ( F Fn A /\ B C_ A ) -> ( F " B ) = { v | E. u e. B ( F ` u ) = v } )
24	23	abeq2d 2191	. . . . 5 |- ( ( F Fn A /\ B C_ A ) -> ( v e. ( F " B ) <-> E. u e. B ( F ` u ) = v ) )
25	16, 24	vtoclg 2658	. . . 4 |- ( C e. _V -> ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) ) )
26	25	impcom 123	. . 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 858	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) )
28		fveq2 5198	. . . 4 |- ( u = x -> ( F ` u ) = ( F ` x ) )
29	28	eqeq1d 2089	. . 3 |- ( u = x -> ( ( F ` u ) = C <-> ( F ` x ) = C ) )
30	29	cbvrexv 2578	. 2 |- ( E. u e. B ( F ` u ) = C <-> E. x e. B ( F ` x ) = C )
31	27, 30	syl6bb 194	1 |- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. x e. B ( F ` x ) = C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067   E.wrex 2349   _Vcvv 2601   C_ wss 2973   dom cdm 4363   "cima 4366   Fun wfun 4916   Fn wfn 4917   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by:  ssimaex  5255  rexima  5415  ralima  5416  f1elima  5433  ovelimab  5671