Theorem fipreima 10175

Description: Given a finite subset A of the range of a function, there exists a finite subset of the domain whose image is A. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
fipreima	|- (((F Fn B /\ B e. M) /\ (A C_ ran F /\ A e. Fin)) -> E.c e. (~PB i^i Fin)(F"c) = A)

Distinct variable groups:   F,c   A,c   B,c   M,c

Proof of Theorem fipreima

Step	Hyp	Ref	Expression
1		simprr 451	. . 3 |- (((F Fn B /\ B e. M) /\ (A C_ ran F /\ A e. Fin)) -> A e. Fin)
2		simplr 449	. . 3 |- (((F Fn B /\ B e. M) /\ (A C_ ran F /\ A e. Fin)) -> B e. M)
3		fvelrnb 4719	. . . . . . . 8 |- (F Fn B -> (x e. ran F <-> E.y e. B (F` y) = x))
4	3	ralbidv 2123	. . . . . . 7 |- (F Fn B -> (A.x e. A x e. ran F <-> A.x e. A E.y e. B (F` y) = x))
5	4	biimpd 170	. . . . . 6 |- (F Fn B -> (A.x e. A x e. ran F -> A.x e. A E.y e. B (F` y) = x))
6		dfss3 2611	. . . . . 6 |- (A C_ ran F <-> A.x e. A x e. ran F)
7	5, 6	syl5ib 223	. . . . 5 |- (F Fn B -> (A C_ ran F -> A.x e. A E.y e. B (F` y) = x))
8	7	imp 377	. . . 4 |- ((F Fn B /\ A C_ ran F) -> A.x e. A E.y e. B (F` y) = x)
9	8	ad2ant2r 445	. . 3 |- (((F Fn B /\ B e. M) /\ (A C_ ran F /\ A e. Fin)) -> A.x e. A E.y e. B (F` y) = x)
10		indexfi 10174	. . 3 |- ((A e. Fin /\ B e. M /\ A.x e. A E.y e. B (F` y) = x) -> E.c e. Fin (c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x))
11	1, 2, 9, 10	syl111anc 1100	. 2 |- (((F Fn B /\ B e. M) /\ (A C_ ran F /\ A e. Fin)) -> E.c e. Fin (c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x))
12		df-pw 3035	. . . . . . . . . . . 12 |- ~PB = {c | c C_ B}
13	12	abeq2i 2001	. . . . . . . . . . 11 |- (c e. ~PB <-> c C_ B)
14	13	biimpri 169	. . . . . . . . . 10 |- (c C_ B -> c e. ~PB)
15	14	anim1i 361	. . . . . . . . 9 |- ((c C_ B /\ c e. Fin) -> (c e. ~PB /\ c e. Fin))
16	15	ancoms 484	. . . . . . . 8 |- ((c e. Fin /\ c C_ B) -> (c e. ~PB /\ c e. Fin))
17		elin 2786	. . . . . . . 8 |- (c e. (~PB i^i Fin) <-> (c e. ~PB /\ c e. Fin))
18	16, 17	sylibr 217	. . . . . . 7 |- ((c e. Fin /\ c C_ B) -> c e. (~PB i^i Fin))
19	18	3ad2antr1 1041	. . . . . 6 |- ((c e. Fin /\ (c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x)) -> c e. (~PB i^i Fin))
20	19	a1i 8	. . . . 5 |- (F Fn B -> ((c e. Fin /\ (c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x)) -> c e. (~PB i^i Fin)))
21		fnfun 4510	. . . . . . . . . . . . 13 |- (F Fn B -> Fun F)
22	21	adantr 425	. . . . . . . . . . . 12 |- ((F Fn B /\ c C_ B) -> Fun F)
23		fndm 4512	. . . . . . . . . . . . . 14 |- (F Fn B -> dom F = B)
24	23	sseq2d 2645	. . . . . . . . . . . . 13 |- (F Fn B -> (c C_ dom F <-> c C_ B))
25	24	biimpar 461	. . . . . . . . . . . 12 |- ((F Fn B /\ c C_ B) -> c C_ dom F)
26		funimass4 4722	. . . . . . . . . . . 12 |- ((Fun F /\ c C_ dom F) -> ((F"c) C_ A <-> A.y e. c (F` y) e. A))
27	22, 25, 26	syl11anc 524	. . . . . . . . . . 11 |- ((F Fn B /\ c C_ B) -> ((F"c) C_ A <-> A.y e. c (F` y) e. A))
28		eleq1a 1966	. . . . . . . . . . . . 13 |- (x e. A -> ((F` y) = x -> (F` y) e. A))
29	28	r19.23aiv 2211	. . . . . . . . . . . 12 |- (E.x e. A (F` y) = x -> (F` y) e. A)
30	29	ralimi 2168	. . . . . . . . . . 11 |- (A.y e. c E.x e. A (F` y) = x -> A.y e. c (F` y) e. A)
31	27, 30	syl5bir 227	. . . . . . . . . 10 |- ((F Fn B /\ c C_ B) -> (A.y e. c E.x e. A (F` y) = x -> (F"c) C_ A))
32	31	impr 422	. . . . . . . . 9 |- ((F Fn B /\ (c C_ B /\ A.y e. c E.x e. A (F` y) = x)) -> (F"c) C_ A)
33	32	3adantr2 1036	. . . . . . . 8 |- ((F Fn B /\ (c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x)) -> (F"c) C_ A)
34		eleq1 1957	. . . . . . . . . . . . . 14 |- ((F` y) = x -> ((F` y) e. (F"c) <-> x e. (F"c)))
35	21	ad2antrr 440	. . . . . . . . . . . . . . 15 |- (((F Fn B /\ c C_ B) /\ y e. c) -> Fun F)
36		simpr 350	. . . . . . . . . . . . . . . . . 18 |- ((Fun F /\ y e. dom F) -> y e. dom F)
37	36	funfni 4513	. . . . . . . . . . . . . . . . 17 |- ((F Fn B /\ y e. B) -> y e. dom F)
38		ssel2 2616	. . . . . . . . . . . . . . . . 17 |- ((c C_ B /\ y e. c) -> y e. B)
39	37, 38	sylan2 500	. . . . . . . . . . . . . . . 16 |- ((F Fn B /\ (c C_ B /\ y e. c)) -> y e. dom F)
40	39	anassrs 489	. . . . . . . . . . . . . . 15 |- (((F Fn B /\ c C_ B) /\ y e. c) -> y e. dom F)
41		simpr 350	. . . . . . . . . . . . . . 15 |- (((F Fn B /\ c C_ B) /\ y e. c) -> y e. c)
42		funfvima 4828	. . . . . . . . . . . . . . . 16 |- ((Fun F /\ y e. dom F) -> (y e. c -> (F` y) e. (F"c)))
43	42	3impia 1064	. . . . . . . . . . . . . . 15 |- ((Fun F /\ y e. dom F /\ y e. c) -> (F` y) e. (F"c))
44	35, 40, 41, 43	syl111anc 1100	. . . . . . . . . . . . . 14 |- (((F Fn B /\ c C_ B) /\ y e. c) -> (F` y) e. (F"c))
45	34, 44	syl5cbi 226	. . . . . . . . . . . . 13 |- (((F Fn B /\ c C_ B) /\ y e. c) -> ((F` y) = x -> x e. (F"c)))
46	45	r19.23adva 2216	. . . . . . . . . . . 12 |- ((F Fn B /\ c C_ B) -> (E.y e. c (F` y) = x -> x e. (F"c)))
47	46	ralimdv 2172	. . . . . . . . . . 11 |- ((F Fn B /\ c C_ B) -> (A.x e. A E.y e. c (F` y) = x -> A.x e. A x e. (F"c)))
48	47	impr 422	. . . . . . . . . 10 |- ((F Fn B /\ (c C_ B /\ A.x e. A E.y e. c (F` y) = x)) -> A.x e. A x e. (F"c))
49		dfss3 2611	. . . . . . . . . 10 |- (A C_ (F"c) <-> A.x e. A x e. (F"c))
50	48, 49	sylibr 217	. . . . . . . . 9 |- ((F Fn B /\ (c C_ B /\ A.x e. A E.y e. c (F` y) = x)) -> A C_ (F"c))
51	50	3adantr3 1037	. . . . . . . 8 |- ((F Fn B /\ (c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x)) -> A C_ (F"c))
52	33, 51	eqssd 2633	. . . . . . 7 |- ((F Fn B /\ (c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x)) -> (F"c) = A)
53	52	ex 402	. . . . . 6 |- (F Fn B -> ((c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x) -> (F"c) = A))
54	53	adantld 426	. . . . 5 |- (F Fn B -> ((c e. Fin /\ (c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x)) -> (F"c) = A))
55	20, 54	jcad 661	. . . 4 |- (F Fn B -> ((c e. Fin /\ (c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x)) -> (c e. (~PB i^i Fin) /\ (F"c) = A)))
56	55	reximdv2 2200	. . 3 |- (F Fn B -> (E.c e. Fin (c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x) -> E.c e. (~PB i^i Fin)(F"c) = A))
57	56	ad2antrr 440	. 2 |- (((F Fn B /\ B e. M) /\ (A C_ ran F /\ A e. Fin)) -> (E.c e. Fin (c C_ B /\ A.x e. A E.y e. c (F` y) = x /\ A.y e. c E.x e. A (F` y) = x) -> E.c e. (~PB i^i Fin)(F"c) = A))
58	11, 57	mpd 29	1 |- (((F Fn B /\ B e. M) /\ (A C_ ran F /\ A e. Fin)) -> E.c e. (~PB i^i Fin)(F"c) = A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593  ~Pcpw 3032  dom cdm 3986  ran crn 3987  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998  Fincfn 5426

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-rep 3428  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-3or 859  df-3an 860  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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-oadd 5179  df-er 5318  df-en 5427  df-fin 5430

Copyright terms: Public domain