Theorem abianfp 5171

Description: "A most fundamental fixed point theorem" of Alexander Abian (1923-1999), apparently proved in 1998. Let G` 0 = x, G` 1 = F` x, G` 2 = F` (F` x),... be the iterates of F. The theorem reads (using our variable names): "Let F be a mapping from a set A into itself. Then F has a fixed point if and only if: There exists an element x of A such that for every ordinal v, G` v is an element of A, and if G` v is not a fixed point of F then the G` u's are all distinct for every ordinal u e. v." See df-rdg 5140 for the rec operation. The proof's key idea is to assume that F does not have a fixed point, then use the Axiom of Replacement in the form of f1dmex 4664 to derive that the class of all ordinal numbers exists, contradicting onprc 3865. Our version of this theorem does not require the hypothesis that F be a mapping. Reference: http://us2.metamath.org:88/abian-themostfixed.html. For an application of this theorem, see http://groups.google.com/group/sci.stat.math/msg/1737ee1133c24aeb for its use in a proof of Tarski's fixed point theorem.

Hypotheses

Ref	Expression
abianfp.1	|- A e. _V
abianfp.2	|- G = rec({<.z, w>. | w = (F` z)}, x)

Assertion

Ref	Expression
abianfp	|- (E.x e. A (F` x) = x <-> E.x e. A A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))))

Distinct variable groups:   x,v,A   x,z,w,u,F,v   v,G,u

Proof of Theorem abianfp

Step	Hyp	Ref	Expression
1		abianfp.1	. . . . . . . . . . 11 |- A e. _V
2		abianfp.2	. . . . . . . . . . 11 |- G = rec({<.z, w>. | w = (F` z)}, x)
3	1, 2	abianfplem 5170	. . . . . . . . . 10 |- (v e. On -> ((F` x) = x -> (G` v) = x))
4	3	imp 377	. . . . . . . . 9 |- ((v e. On /\ (F` x) = x) -> (G` v) = x)
5	4	eleq1d 1963	. . . . . . . 8 |- ((v e. On /\ (F` x) = x) -> ((G` v) e. A <-> x e. A))
6	5	biimprd 171	. . . . . . 7 |- ((v e. On /\ (F` x) = x) -> (x e. A -> (G` v) e. A))
7		fveq2 4681	. . . . . . . . . . . 12 |- ((G` v) = x -> (F` (G` v)) = (F` x))
8		id 73	. . . . . . . . . . . 12 |- ((G` v) = x -> (G` v) = x)
9	7, 8	eqeq12d 1899	. . . . . . . . . . 11 |- ((G` v) = x -> ((F` (G` v)) = (G` v) <-> (F` x) = x))
10	9	biimprcd 173	. . . . . . . . . 10 |- ((F` x) = x -> ((G` v) = x -> (F` (G` v)) = (G` v)))
11	3, 10	sylcom 62	. . . . . . . . 9 |- (v e. On -> ((F` x) = x -> (F` (G` v)) = (G` v)))
12	11	imp 377	. . . . . . . 8 |- ((v e. On /\ (F` x) = x) -> (F` (G` v)) = (G` v))
13	12	pm2.24d 120	. . . . . . 7 |- ((v e. On /\ (F` x) = x) -> (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u)))
14	6, 13	jctird 663	. . . . . 6 |- ((v e. On /\ (F` x) = x) -> (x e. A -> ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u)))))
15	14	ex 402	. . . . 5 |- (v e. On -> ((F` x) = x -> (x e. A -> ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))))))
16	15	com13 37	. . . 4 |- (x e. A -> ((F` x) = x -> (v e. On -> ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))))))
17	16	r19.21adv 2181	. . 3 |- (x e. A -> ((F` x) = x -> A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u)))))
18	17	reximia 2196	. 2 |- (E.x e. A (F` x) = x -> E.x e. A A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))))
19		onprc 3865	. . . . . 6 |- -. On e. _V
20		r19.26 2219	. . . . . . 7 |- (A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))) <-> (A.v e. On (G` v) e. A /\ A.v e. On (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))))
21		fveq2 4681	. . . . . . . . . . . . . . . . . . 19 |- (y = (G` v) -> (F` y) = (F` (G` v)))
22		id 73	. . . . . . . . . . . . . . . . . . 19 |- (y = (G` v) -> y = (G` v))
23	21, 22	eqeq12d 1899	. . . . . . . . . . . . . . . . . 18 |- (y = (G` v) -> ((F` y) = y <-> (F` (G` v)) = (G` v)))
24	23	notbid 673	. . . . . . . . . . . . . . . . 17 |- (y = (G` v) -> (-. (F` y) = y <-> -. (F` (G` v)) = (G` v)))
25	24	rcla4cv 2377	. . . . . . . . . . . . . . . 16 |- (A.y e. A -. (F` y) = y -> ((G` v) e. A -> -. (F` (G` v)) = (G` v)))
26	25	imim1d 33	. . . . . . . . . . . . . . 15 |- (A.y e. A -. (F` y) = y -> ((-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u)) -> ((G` v) e. A -> A.u e. v -. (G` v) = (G` u))))
27	26	ralimdv 2172	. . . . . . . . . . . . . 14 |- (A.y e. A -. (F` y) = y -> (A.v e. On (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u)) -> A.v e. On ((G` v) e. A -> A.u e. v -. (G` v) = (G` u))))
28		ralim 2164	. . . . . . . . . . . . . 14 |- (A.v e. On ((G` v) e. A -> A.u e. v -. (G` v) = (G` u)) -> (A.v e. On (G` v) e. A -> A.v e. On A.u e. v -. (G` v) = (G` u)))
29	27, 28	syl6 25	. . . . . . . . . . . . 13 |- (A.y e. A -. (F` y) = y -> (A.v e. On (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u)) -> (A.v e. On (G` v) e. A -> A.v e. On A.u e. v -. (G` v) = (G` u))))
30	29	imp 377	. . . . . . . . . . . 12 |- ((A.y e. A -. (F` y) = y /\ A.v e. On (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))) -> (A.v e. On (G` v) e. A -> A.v e. On A.u e. v -. (G` v) = (G` u)))
31	30	com12 14	. . . . . . . . . . 11 |- (A.v e. On (G` v) e. A -> ((A.y e. A -. (F` y) = y /\ A.v e. On (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))) -> A.v e. On A.u e. v -. (G` v) = (G` u)))
32		f1dmex 4664	. . . . . . . . . . . . 13 |- ((G:On-1-1->A /\ A e. _V) -> On e. _V)
33		rdgfnon 5147	. . . . . . . . . . . . . . . . 17 |- rec({<.z, w>. | w = (F` z)}, x) Fn On
34	2	fneq1i 4507	. . . . . . . . . . . . . . . . 17 |- (G Fn On <-> rec({<.z, w>. | w = (F` z)}, x) Fn On)
35	33, 34	mpbir 207	. . . . . . . . . . . . . . . 16 |- G Fn On
36		ffnfv 4801	. . . . . . . . . . . . . . . . 17 |- (G:On-->A <-> (G Fn On /\ A.v e. On (G` v) e. A))
37	36	biimpri 169	. . . . . . . . . . . . . . . 16 |- ((G Fn On /\ A.v e. On (G` v) e. A) -> G:On-->A)
38	35, 37	mpan 759	. . . . . . . . . . . . . . 15 |- (A.v e. On (G` v) e. A -> G:On-->A)
39		ssid 2634	. . . . . . . . . . . . . . . . 17 |- On C_ On
40	35	tz7.48lem 5164	. . . . . . . . . . . . . . . . 17 |- ((On C_ On /\ A.v e. On A.u e. v -. (G` v) = (G` u)) -> Fun `'(G |` On))
41	39, 40	mpan 759	. . . . . . . . . . . . . . . 16 |- (A.v e. On A.u e. v -. (G` v) = (G` u) -> Fun `'(G |` On))
42		fnresdm 4522	. . . . . . . . . . . . . . . . . . 19 |- (G Fn On -> (G |` On) = G)
43	35, 42	ax-mp 7	. . . . . . . . . . . . . . . . . 18 |- (G |` On) = G
44	43	cnveqi 4136	. . . . . . . . . . . . . . . . 17 |- `'(G |` On) = `'G
45		funeq 4441	. . . . . . . . . . . . . . . . 17 |- (`'(G |` On) = `'G -> (Fun `'(G |` On) <-> Fun `'G))
46	44, 45	ax-mp 7	. . . . . . . . . . . . . . . 16 |- (Fun `'(G |` On) <-> Fun `'G)
47	41, 46	sylib 215	. . . . . . . . . . . . . . 15 |- (A.v e. On A.u e. v -. (G` v) = (G` u) -> Fun `'G)
48	38, 47	anim12i 360	. . . . . . . . . . . . . 14 |- ((A.v e. On (G` v) e. A /\ A.v e. On A.u e. v -. (G` v) = (G` u)) -> (G:On-->A /\ Fun `'G))
49		df-f1 4011	. . . . . . . . . . . . . 14 |- (G:On-1-1->A <-> (G:On-->A /\ Fun `'G))
50	48, 49	sylibr 217	. . . . . . . . . . . . 13 |- ((A.v e. On (G` v) e. A /\ A.v e. On A.u e. v -. (G` v) = (G` u)) -> G:On-1-1->A)
51	32, 50, 1	sylancl 525	. . . . . . . . . . . 12 |- ((A.v e. On (G` v) e. A /\ A.v e. On A.u e. v -. (G` v) = (G` u)) -> On e. _V)
52	51	ex 402	. . . . . . . . . . 11 |- (A.v e. On (G` v) e. A -> (A.v e. On A.u e. v -. (G` v) = (G` u) -> On e. _V))
53	31, 52	syld 30	. . . . . . . . . 10 |- (A.v e. On (G` v) e. A -> ((A.y e. A -. (F` y) = y /\ A.v e. On (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))) -> On e. _V))
54	53	exp3a 405	. . . . . . . . 9 |- (A.v e. On (G` v) e. A -> (A.y e. A -. (F` y) = y -> (A.v e. On (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u)) -> On e. _V)))
55	54	com23 36	. . . . . . . 8 |- (A.v e. On (G` v) e. A -> (A.v e. On (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u)) -> (A.y e. A -. (F` y) = y -> On e. _V)))
56	55	imp 377	. . . . . . 7 |- ((A.v e. On (G` v) e. A /\ A.v e. On (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))) -> (A.y e. A -. (F` y) = y -> On e. _V))
57	20, 56	sylbi 216	. . . . . 6 |- (A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))) -> (A.y e. A -. (F` y) = y -> On e. _V))
58	19, 57	mtoi 122	. . . . 5 |- (A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))) -> -. A.y e. A -. (F` y) = y)
59	58	a1i 8	. . . 4 |- (x e. A -> (A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))) -> -. A.y e. A -. (F` y) = y))
60	59	r19.23aiv 2211	. . 3 |- (E.x e. A A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))) -> -. A.y e. A -. (F` y) = y)
61		fveq2 4681	. . . . . 6 |- (x = y -> (F` x) = (F` y))
62		id 73	. . . . . 6 |- (x = y -> x = y)
63	61, 62	eqeq12d 1899	. . . . 5 |- (x = y -> ((F` x) = x <-> (F` y) = y))
64	63	cbvrexv 2281	. . . 4 |- (E.x e. A (F` x) = x <-> E.y e. A (F` y) = y)
65		dfrex2 2116	. . . 4 |- (E.y e. A (F` y) = y <-> -. A.y e. A -. (F` y) = y)
66	64, 65	bitr2i 191	. . 3 |- (-. A.y e. A -. (F` y) = y <-> E.x e. A (F` x) = x)
67	60, 66	sylib 215	. 2 |- (E.x e. A A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))) -> E.x e. A (F` x) = x)
68	18, 67	impbii 174	1 |- (E.x e. A (F` x) = x <-> E.x e. A A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  {copab 3395  Oncon0 3657  `'ccnv 3985   |` cres 3988  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  ` cfv 3998  reccrdg 5139

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-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-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-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-rdg 5140

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