Theorem fvelimab 4725

Description: Function value in an image. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (An unnecessary distinct variable restriction was removed 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

Proof of Theorem fvelimab

Step	Hyp	Ref	Expression
1		elisset 2299	. . 3 |- (C e. (F"B) -> C e. _V)
2	1	anim2i 362	. 2 |- (((F Fn A /\ B C_ A) /\ C e. (F"B)) -> ((F Fn A /\ B C_ A) /\ C e. _V))
3		fvex 4689	. . . . . 6 |- (F` x) e. _V
4		eleq1 1957	. . . . . 6 |- ((F` x) = C -> ((F` x) e. _V <-> C e. _V))
5	3, 4	mpbii 210	. . . . 5 |- ((F` x) = C -> C e. _V)
6	5	a1i 8	. . . 4 |- (x e. B -> ((F` x) = C -> C e. _V))
7	6	r19.23aiv 2211	. . 3 |- (E.x e. B (F` x) = C -> C e. _V)
8	7	anim2i 362	. 2 |- (((F Fn A /\ B C_ A) /\ E.x e. B (F` x) = C) -> ((F Fn A /\ B C_ A) /\ C e. _V))
9		eleq1 1957	. . . . . 6 |- (y = C -> (y e. (F"B) <-> C e. (F"B)))
10		eqeq2 1893	. . . . . . 7 |- (y = C -> ((F` x) = y <-> (F` x) = C))
11	10	rexbidv 2124	. . . . . 6 |- (y = C -> (E.x e. B (F` x) = y <-> E.x e. B (F` x) = C))
12	9, 11	bibi12d 691	. . . . 5 |- (y = C -> ((y e. (F"B) <-> E.x e. B (F` x) = y) <-> (C e. (F"B) <-> E.x e. B (F` x) = C)))
13	12	imbi2d 674	. . . 4 |- (y = C -> (((F Fn A /\ B C_ A) -> (y e. (F"B) <-> E.x e. B (F` x) = y)) <-> ((F Fn A /\ B C_ A) -> (C e. (F"B) <-> E.x e. B (F` x) = C))))
14		fnfun 4510	. . . . . . 7 |- (F Fn A -> Fun F)
15	14	adantr 425	. . . . . 6 |- ((F Fn A /\ B C_ A) -> Fun F)
16		fndm 4512	. . . . . . . 8 |- (F Fn A -> dom F = A)
17	16	sseq2d 2645	. . . . . . 7 |- (F Fn A -> (B C_ dom F <-> B C_ A))
18	17	biimpar 461	. . . . . 6 |- ((F Fn A /\ B C_ A) -> B C_ dom F)
19		dfimafn 4720	. . . . . 6 |- ((Fun F /\ B C_ dom F) -> (F"B) = {y | E.x e. B (F` x) = y})
20	15, 18, 19	syl11anc 524	. . . . 5 |- ((F Fn A /\ B C_ A) -> (F"B) = {y | E.x e. B (F` x) = y})
21	20	abeq2d 2003	. . . 4 |- ((F Fn A /\ B C_ A) -> (y e. (F"B) <-> E.x e. B (F` x) = y))
22	13, 21	vtoclg 2346	. . 3 |- (C e. _V -> ((F Fn A /\ B C_ A) -> (C e. (F"B) <-> E.x e. B (F` x) = C)))
23	22	impcom 378	. 2 |- (((F Fn A /\ B C_ A) /\ C e. _V) -> (C e. (F"B) <-> E.x e. B (F` x) = C))
24	2, 8, 23	pm5.21nd 744	1 |- ((F Fn A /\ B C_ A) -> (C e. (F"B) <-> E.x e. B (F` x) = C))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  _Vcvv 2292   C_ wss 2593  dom cdm 3986  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  ssimaex 4729  ac6sfilem3 5508  pjimai 11748  nocvxmin 14029  npincppr 14501  filnetlem5 15644  raleqfn 15672  f1elima 15719  ismtyhmeolem 15950  heiborlem10 15964  heiborlem11 15965

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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