Theorem abianfp 3000

Description: "A most fundamental fixed point theorem" of Alexander Abian (1923-1999), apparently proved in 1998. "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 ∈ v." Note that G ‘0 = x, G ‘1 = F ‘x, G ‘2 = F ‘(F ‘x),... are the iterates of F. See df-rdg 2970 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 2819 to derive that the class of all ordinals exists, contradicting onprc 2240. Our version of this theorem does not require the hypothesis that F be a mapping. Reference: http://www.math.ucdavis.edu/~suh/abian/abian-themostfixed.html.

Hypotheses

Ref	Expression
abianfp.1	⊢ A ∈ V
abianfp.2	⊢ G = rec({⟨z, w⟩∣w = (F ‘z)}, x)

Assertion

Ref	Expression
abianfp	⊢ (∃x ∈ A (F ‘x) = x ↔ ∃x ∈ A ∀v ∈ On ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))))

Distinct variable group(s):   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 ∈ V
2		abianfp.2	. . . . . . . . . . 11 ⊢ G = rec({⟨z, w⟩∣w = (F ‘z)}, x)
3	1, 2	abianfplem 2999	. . . . . . . . . 10 ⊢ (v ∈ On → ((F ‘x) = x → (G ‘v) = x))
4	3	imp 277	. . . . . . . . 9 ⊢ ((v ∈ On ∧ (F ‘x) = x) → (G ‘v) = x)
5	4	eleq1d 1155	. . . . . . . 8 ⊢ ((v ∈ On ∧ (F ‘x) = x) → ((G ‘v) ∈ A ↔ x ∈ A))
6	5	biimprd 136	. . . . . . 7 ⊢ ((v ∈ On ∧ (F ‘x) = x) → (x ∈ A → (G ‘v) ∈ A))
7		fveq2 2832	. . . . . . . . . . . . . 14 ⊢ ((G ‘v) = x → (F ‘(G ‘v)) = (F ‘x))
8		id 9	. . . . . . . . . . . . . 14 ⊢ ((G ‘v) = x → (G ‘v) = x)
9	7, 8	cleq12d 1115	. . . . . . . . . . . . 13 ⊢ ((G ‘v) = x → ((F ‘(G ‘v)) = (G ‘v) ↔ (F ‘x) = x))
10	9	biimprcd 138	. . . . . . . . . . . 12 ⊢ ((F ‘x) = x → ((G ‘v) = x → (F ‘(G ‘v)) = (G ‘v)))
11	3, 10	sylcom 51	. . . . . . . . . . 11 ⊢ (v ∈ On → ((F ‘x) = x → (F ‘(G ‘v)) = (G ‘v)))
12	11	imp 277	. . . . . . . . . 10 ⊢ ((v ∈ On ∧ (F ‘x) = x) → (F ‘(G ‘v)) = (G ‘v))
13		negb 79	. . . . . . . . . 10 ⊢ ((F ‘(G ‘v)) = (G ‘v) → ¬ ¬ (F ‘(G ‘v)) = (G ‘v))
14	12, 13	syl 12	. . . . . . . . 9 ⊢ ((v ∈ On ∧ (F ‘x) = x) → ¬ ¬ (F ‘(G ‘v)) = (G ‘v))
15	14	pm2.21d 74	. . . . . . . 8 ⊢ ((v ∈ On ∧ (F ‘x) = x) → (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u)))
16	15	a1d 14	. . . . . . 7 ⊢ ((v ∈ On ∧ (F ‘x) = x) → (x ∈ A → (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))))
17	6, 16	jcad 455	. . . . . 6 ⊢ ((v ∈ On ∧ (F ‘x) = x) → (x ∈ A → ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u)))))
18	17	exp 291	. . . . 5 ⊢ (v ∈ On → ((F ‘x) = x → (x ∈ A → ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))))))
19	18	com13 33	. . . 4 ⊢ (x ∈ A → ((F ‘x) = x → (v ∈ On → ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))))))
20	19	r19.21adv 1262	. . 3 ⊢ (x ∈ A → ((F ‘x) = x → ∀v ∈ On ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u)))))
21	20	r19.22i 1273	. 2 ⊢ (∃x ∈ A (F ‘x) = x → ∃x ∈ A ∀v ∈ On ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))))
22		onprc 2240	. . . . . 6 ⊢ ¬ On ∈ V
23		r19.26 1289	. . . . . . 7 ⊢ (∀v ∈ On ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))) ↔ (∀v ∈ On (G ‘v) ∈ A ∧ ∀v ∈ On (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))))
24		fveq2 2832	. . . . . . . . . . . . . . . . . . 19 ⊢ (y = (G ‘v) → (F ‘y) = (F ‘(G ‘v)))
25		id 9	. . . . . . . . . . . . . . . . . . 19 ⊢ (y = (G ‘v) → y = (G ‘v))
26	24, 25	cleq12d 1115	. . . . . . . . . . . . . . . . . 18 ⊢ (y = (G ‘v) → ((F ‘y) = y ↔ (F ‘(G ‘v)) = (G ‘v)))
27	26	negbid 463	. . . . . . . . . . . . . . . . 17 ⊢ (y = (G ‘v) → (¬ (F ‘y) = y ↔ ¬ (F ‘(G ‘v)) = (G ‘v)))
28	27	rcla4v 1402	. . . . . . . . . . . . . . . 16 ⊢ (∀y ∈ A ¬ (F ‘y) = y → ((G ‘v) ∈ A → ¬ (F ‘(G ‘v)) = (G ‘v)))
29	28	syl4d 28	. . . . . . . . . . . . . . 15 ⊢ (∀y ∈ A ¬ (F ‘y) = y → ((¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u)) → ((G ‘v) ∈ A → ∀u ∈ v ¬ (G ‘v) = (G ‘u))))
30	29	r19.20sdv 1257	. . . . . . . . . . . . . 14 ⊢ (∀y ∈ A ¬ (F ‘y) = y → (∀v ∈ On (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u)) → ∀v ∈ On ((G ‘v) ∈ A → ∀u ∈ v ¬ (G ‘v) = (G ‘u))))
31		r19.20 1251	. . . . . . . . . . . . . 14 ⊢ (∀v ∈ On ((G ‘v) ∈ A → ∀u ∈ v ¬ (G ‘v) = (G ‘u)) → (∀v ∈ On (G ‘v) ∈ A → ∀v ∈ On ∀u ∈ v ¬ (G ‘v) = (G ‘u)))
32	30, 31	syl6 23	. . . . . . . . . . . . 13 ⊢ (∀y ∈ A ¬ (F ‘y) = y → (∀v ∈ On (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u)) → (∀v ∈ On (G ‘v) ∈ A → ∀v ∈ On ∀u ∈ v ¬ (G ‘v) = (G ‘u))))
33	32	imp 277	. . . . . . . . . . . 12 ⊢ ((∀y ∈ A ¬ (F ‘y) = y ∧ ∀v ∈ On (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))) → (∀v ∈ On (G ‘v) ∈ A → ∀v ∈ On ∀u ∈ v ¬ (G ‘v) = (G ‘u)))
34	33	com12 13	. . . . . . . . . . 11 ⊢ (∀v ∈ On (G ‘v) ∈ A → ((∀y ∈ A ¬ (F ‘y) = y ∧ ∀v ∈ On (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))) → ∀v ∈ On ∀u ∈ v ¬ (G ‘v) = (G ‘u)))
35		rdgfnon 2977	. . . . . . . . . . . . . . . . 17 ⊢ rec({⟨z, w⟩∣w = (F ‘z)}, x) Fn On
36		fneq1 2718	. . . . . . . . . . . . . . . . . 18 ⊢ (G = rec({⟨z, w⟩∣w = (F ‘z)}, x) → (G Fn On ↔ rec({⟨z, w⟩∣w = (F ‘z)}, x) Fn On))
37	2, 36	ax-mp 6	. . . . . . . . . . . . . . . . 17 ⊢ (G Fn On ↔ rec({⟨z, w⟩∣w = (F ‘z)}, x) Fn On)
38	35, 37	mpbir 165	. . . . . . . . . . . . . . . 16 ⊢ G Fn On
39		ffnfv 2892	. . . . . . . . . . . . . . . . 17 ⊢ (G:On–→A ↔ (G Fn On ∧ ∀v ∈ On (G ‘v) ∈ A))
40	39	biimpr 134	. . . . . . . . . . . . . . . 16 ⊢ ((G Fn On ∧ ∀v ∈ On (G ‘v) ∈ A) → G:On–→A)
41	38, 40	mpan 518	. . . . . . . . . . . . . . 15 ⊢ (∀v ∈ On (G ‘v) ∈ A → G:On–→A)
42		ssid 1519	. . . . . . . . . . . . . . . . 17 ⊢ On ⊆ On
43	38	tz7.48lem 2993	. . . . . . . . . . . . . . . . 17 ⊢ ((On ⊆ On ∧ ∀v ∈ On ∀u ∈ v ¬ (G ‘v) = (G ‘u)) → Fun ◡(G ↾ On))
44	42, 43	mpan 518	. . . . . . . . . . . . . . . 16 ⊢ (∀v ∈ On ∀u ∈ v ¬ (G ‘v) = (G ‘u) → Fun ◡(G ↾ On))
45		fnresdm 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ (G Fn On → (G ↾ On) = G)
46	38, 45	ax-mp 6	. . . . . . . . . . . . . . . . . 18 ⊢ (G ↾ On) = G
47		cnveq 2513	. . . . . . . . . . . . . . . . . 18 ⊢ ((G ↾ On) = G → ◡(G ↾ On) = ◡G)
48	46, 47	ax-mp 6	. . . . . . . . . . . . . . . . 17 ⊢ ◡(G ↾ On) = ◡G
49		funeq 2683	. . . . . . . . . . . . . . . . 17 ⊢ (◡(G ↾ On) = ◡G → (Fun ◡(G ↾ On) ↔ Fun ◡G))
50	48, 49	ax-mp 6	. . . . . . . . . . . . . . . 16 ⊢ (Fun ◡(G ↾ On) ↔ Fun ◡G)
51	44, 50	sylib 173	. . . . . . . . . . . . . . 15 ⊢ (∀v ∈ On ∀u ∈ v ¬ (G ‘v) = (G ‘u) → Fun ◡G)
52	41, 51	anim12i 268	. . . . . . . . . . . . . 14 ⊢ ((∀v ∈ On (G ‘v) ∈ A ∧ ∀v ∈ On ∀u ∈ v ¬ (G ‘v) = (G ‘u)) → (G:On–→A ∧ Fun ◡G))
53		df-f1 2435	. . . . . . . . . . . . . 14 ⊢ (G:On–1-1→A ↔ (G:On–→A ∧ Fun ◡G))
54	52, 53	sylibr 175	. . . . . . . . . . . . 13 ⊢ ((∀v ∈ On (G ‘v) ∈ A ∧ ∀v ∈ On ∀u ∈ v ¬ (G ‘v) = (G ‘u)) → G:On–1-1→A)
55		f1dmex 2819	. . . . . . . . . . . . . 14 ⊢ (A ∈ V → (G:On–1-1→A → On ∈ V))
56	1, 55	ax-mp 6	. . . . . . . . . . . . 13 ⊢ (G:On–1-1→A → On ∈ V)
57	54, 56	syl 12	. . . . . . . . . . . 12 ⊢ ((∀v ∈ On (G ‘v) ∈ A ∧ ∀v ∈ On ∀u ∈ v ¬ (G ‘v) = (G ‘u)) → On ∈ V)
58	57	exp 291	. . . . . . . . . . 11 ⊢ (∀v ∈ On (G ‘v) ∈ A → (∀v ∈ On ∀u ∈ v ¬ (G ‘v) = (G ‘u) → On ∈ V))
59	34, 58	syld 27	. . . . . . . . . 10 ⊢ (∀v ∈ On (G ‘v) ∈ A → ((∀y ∈ A ¬ (F ‘y) = y ∧ ∀v ∈ On (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))) → On ∈ V))
60	59	exp3a 292	. . . . . . . . 9 ⊢ (∀v ∈ On (G ‘v) ∈ A → (∀y ∈ A ¬ (F ‘y) = y → (∀v ∈ On (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u)) → On ∈ V)))
61	60	com23 32	. . . . . . . 8 ⊢ (∀v ∈ On (G ‘v) ∈ A → (∀v ∈ On (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u)) → (∀y ∈ A ¬ (F ‘y) = y → On ∈ V)))
62	61	imp 277	. . . . . . 7 ⊢ ((∀v ∈ On (G ‘v) ∈ A ∧ ∀v ∈ On (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))) → (∀y ∈ A ¬ (F ‘y) = y → On ∈ V))
63	23, 62	sylbi 174	. . . . . 6 ⊢ (∀v ∈ On ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))) → (∀y ∈ A ¬ (F ‘y) = y → On ∈ V))
64	22, 63	mtoi 94	. . . . 5 ⊢ (∀v ∈ On ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))) → ¬ ∀y ∈ A ¬ (F ‘y) = y)
65	64	a1i 7	. . . 4 ⊢ (x ∈ A → (∀v ∈ On ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))) → ¬ ∀y ∈ A ¬ (F ‘y) = y))
66	65	r19.23aiv 1284	. . 3 ⊢ (∃x ∈ A ∀v ∈ On ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))) → ¬ ∀y ∈ A ¬ (F ‘y) = y)
67		fveq2 2832	. . . . . 6 ⊢ (x = y → (F ‘x) = (F ‘y))
68		id 9	. . . . . 6 ⊢ (x = y → x = y)
69	67, 68	cleq12d 1115	. . . . 5 ⊢ (x = y → ((F ‘x) = x ↔ (F ‘y) = y))
70	69	cbvrexv 1334	. . . 4 ⊢ (∃x ∈ A (F ‘x) = x ↔ ∃y ∈ A (F ‘y) = y)
71		dfrex2 1212	. . . 4 ⊢ (∃y ∈ A (F ‘y) = y ↔ ¬ ∀y ∈ A ¬ (F ‘y) = y)
72	70, 71	bitr2 152	. . 3 ⊢ (¬ ∀y ∈ A ¬ (F ‘y) = y ↔ ∃x ∈ A (F ‘x) = x)
73	66, 72	sylib 173	. 2 ⊢ (∃x ∈ A ∀v ∈ On ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))) → ∃x ∈ A (F ‘x) = x)
74	21, 73	impbi 139	1 ⊢ (∃x ∈ A (F ‘x) = x ↔ ∃x ∈ A ∀v ∈ On ((G ‘v) ∈ A ∧ (¬ (F ‘(G ‘v)) = (G ‘v) → ∀u ∈ v ¬ (G ‘v) = (G ‘u))))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  {copab 2055  Oncon0 2199  ^◡ccnv 2409   ↾ cres 2412  Fun wfun 2416   Fn wfn 2417  –→wf 2418  –1-1→wf1 2419   ‘cfv 2422  reccrdg 2969

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970

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