Theorem funimass4 5167

Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
funimass4	|- (( Fun F /\ A C_ dom F) -> (( F "A) C_ B <-> A.x e. A ( F ` x) e. B))

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

Proof of Theorem funimass4

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 2928	. 2 |- (( F "A) C_ B <-> A.y(y e. ( F "A) -> y e. B))
2		eqcom 2039	. . . . . . . . . 10 |- (y = ( F ` x) <-> ( F ` x) = y)
3		ssel 2933	. . . . . . . . . . . 12 |- (A C_ dom F -> (x e. A -> x e. dom F))
4		funbrfvb 5159	. . . . . . . . . . . . 13 |- (( Fun F /\ x e. dom F) -> (( F ` x) = y <-> x Fy))
5	4	ex 108	. . . . . . . . . . . 12 |- ( Fun F -> (x e. dom F -> (( F ` x) = y <-> x Fy)))
6	3, 5	syl9 66	. . . . . . . . . . 11 |- (A C_ dom F -> ( Fun F -> (x e. A -> (( F ` x) = y <-> x Fy))))
7	6	imp31 243	. . . . . . . . . 10 |- (((A C_ dom F /\ Fun F) /\ x e. A) -> (( F ` x) = y <-> x Fy))
8	2, 7	syl5bb 181	. . . . . . . . 9 |- (((A C_ dom F /\ Fun F) /\ x e. A) -> (y = ( F ` x) <-> x Fy))
9	8	rexbidva 2317	. . . . . . . 8 |- ((A C_ dom F /\ Fun F) -> (E.x e. A y = ( F ` x) <-> E.x e. A x Fy))
10		vex 2554	. . . . . . . . 9 |- y e. _V
11	10	elima 4616	. . . . . . . 8 |- (y e. ( F "A) <-> E.x e. A x Fy)
12	9, 11	syl6rbbr 188	. . . . . . 7 |- ((A C_ dom F /\ Fun F) -> (y e. ( F "A) <-> E.x e. A y = ( F ` x)))
13	12	imbi1d 220	. . . . . 6 |- ((A C_ dom F /\ Fun F) -> ((y e. ( F "A) -> y e. B) <-> (E.x e. A y = ( F ` x) -> y e. B)))
14		r19.23v 2419	. . . . . 6 |- (A.x e. A (y = ( F ` x) -> y e. B) <-> (E.x e. A y = ( F ` x) -> y e. B))
15	13, 14	syl6bbr 187	. . . . 5 |- ((A C_ dom F /\ Fun F) -> ((y e. ( F "A) -> y e. B) <-> A.x e. A (y = ( F ` x) -> y e. B)))
16	15	albidv 1702	. . . 4 |- ((A C_ dom F /\ Fun F) -> (A.y(y e. ( F "A) -> y e. B) <-> A.yA.x e. A (y = ( F ` x) -> y e. B)))
17	16	ancoms 255	. . 3 |- (( Fun F /\ A C_ dom F) -> (A.y(y e. ( F "A) -> y e. B) <-> A.yA.x e. A (y = ( F ` x) -> y e. B)))
18		ralcom4 2570	. . . 4 |- (A.x e. A A.y(y = ( F ` x) -> y e. B) <-> A.yA.x e. A (y = ( F ` x) -> y e. B))
19		ssel2 2934	. . . . . . . . 9 |- ((A C_ dom F /\ x e. A) -> x e. dom F)
20	19	anim2i 324	. . . . . . . 8 |- (( Fun F /\ (A C_ dom F /\ x e. A)) -> ( Fun F /\ x e. dom F))
21	20	3impb 1099	. . . . . . 7 |- (( Fun F /\ A C_ dom F /\ x e. A) -> ( Fun F /\ x e. dom F))
22		funfvex 5135	. . . . . . 7 |- (( Fun F /\ x e. dom F) -> ( F ` x) e. _V)
23		nfv 1418	. . . . . . . 8 |- F/y( F ` x) e. B
24		eleq1 2097	. . . . . . . 8 |- (y = ( F ` x) -> (y e. B <-> ( F ` x) e. B))
25	23, 24	ceqsalg 2576	. . . . . . 7 |- (( F ` x) e. _V -> (A.y(y = ( F ` x) -> y e. B) <-> ( F ` x) e. B))
26	21, 22, 25	3syl 17	. . . . . 6 |- (( Fun F /\ A C_ dom F /\ x e. A) -> (A.y(y = ( F ` x) -> y e. B) <-> ( F ` x) e. B))
27	26	3expa 1103	. . . . 5 |- ((( Fun F /\ A C_ dom F) /\ x e. A) -> (A.y(y = ( F ` x) -> y e. B) <-> ( F ` x) e. B))
28	27	ralbidva 2316	. . . 4 |- (( Fun F /\ A C_ dom F) -> (A.x e. A A.y(y = ( F ` x) -> y e. B) <-> A.x e. A ( F ` x) e. B))
29	18, 28	syl5bbr 183	. . 3 |- (( Fun F /\ A C_ dom F) -> (A.yA.x e. A (y = ( F ` x) -> y e. B) <-> A.x e. A ( F ` x) e. B))
30	17, 29	bitrd 177	. 2 |- (( Fun F /\ A C_ dom F) -> (A.y(y e. ( F "A) -> y e. B) <-> A.x e. A ( F ` x) e. B))
31	1, 30	syl5bb 181	1 |- (( Fun F /\ A C_ dom F) -> (( F "A) C_ B <-> A.x e. A ( F ` x) e. B))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884  A.wal 1240   = wceq 1242   e. wcel 1390  A.wral 2300  E.wrex 2301   _Vcvv 2551   C_ wss 2911   class class class wbr 3755   dom cdm 4288   "cima 4291   Fun wfun 4839   ` cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by:  funimass3  5226  funimass5  5227  funconstss  5228  funimassov  5592