Theorem abianfplem 2999

Description: Lemma for abianfp 3000. We prove by transfinite induction that if F has a fixed point x, then its iterates also equal x. This lemma is used for the "trivial" direction of the main theorem.

Hypotheses

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

Assertion

Ref	Expression
abianfplem	⊢ (v ∈ On → ((F ‘x) = x → (G ‘v) = x))

Distinct variable group(s):   x,v,A   x,z,w,F,v   v,G

Proof of Theorem abianfplem

Step	Hyp	Ref	Expression
1		fveq2 2832	. . 3 ⊢ (v = ∅ → (G ‘v) = (G ‘∅))
2	1	cleq1d 1109	. 2 ⊢ (v = ∅ → ((G ‘v) = x ↔ (G ‘∅) = x))
3		fveq2 2832	. . 3 ⊢ (v = y → (G ‘v) = (G ‘y))
4	3	cleq1d 1109	. 2 ⊢ (v = y → ((G ‘v) = x ↔ (G ‘y) = x))
5		fveq2 2832	. . 3 ⊢ (v = suc y → (G ‘v) = (G ‘suc y))
6	5	cleq1d 1109	. 2 ⊢ (v = suc y → ((G ‘v) = x ↔ (G ‘suc y) = x))
7		abianfp.2	. . . . 5 ⊢ G = rec({⟨z, w⟩∣w = (F ‘z)}, x)
8	7	fveq1i 2833	. . . 4 ⊢ (G ‘∅) = (rec({⟨z, w⟩∣w = (F ‘z)}, x) ‘∅)
9		visset 1350	. . . . 5 ⊢ x ∈ V
10	9	rdgzer 2979	. . . 4 ⊢ (rec({⟨z, w⟩∣w = (F ‘z)}, x) ‘∅) = x
11	8, 10	eqtr 1119	. . 3 ⊢ (G ‘∅) = x
12	11	a1i 7	. 2 ⊢ ((F ‘x) = x → (G ‘∅) = x)
13		fvex 2838	. . . . 5 ⊢ (F ‘(G ‘y)) ∈ V
14		ax-17 925	. . . . . 6 ⊢ (u ∈ x → ∀z u ∈ x)
15		ax-17 925	. . . . . 6 ⊢ (u ∈ y → ∀z u ∈ y)
16		ax-17 925	. . . . . . 7 ⊢ (u ∈ F → ∀z u ∈ F)
17		hbopab1 2112	. . . . . . . . . 10 ⊢ (u ∈ {⟨z, w⟩∣w = (F ‘z)} → ∀z u ∈ {⟨z, w⟩∣w = (F ‘z)})
18	17, 14	hbrdg 2974	. . . . . . . . 9 ⊢ (u ∈ rec({⟨z, w⟩∣w = (F ‘z)}, x) → ∀z u ∈ rec({⟨z, w⟩∣w = (F ‘z)}, x))
19	7	eleq2i 1153	. . . . . . . . 9 ⊢ (u ∈ G ↔ u ∈ rec({⟨z, w⟩∣w = (F ‘z)}, x))
20	19	bial 695	. . . . . . . . 9 ⊢ (∀z u ∈ G ↔ ∀z u ∈ rec({⟨z, w⟩∣w = (F ‘z)}, x))
21	18, 19, 20	3imtr4 192	. . . . . . . 8 ⊢ (u ∈ G → ∀z u ∈ G)
22	21, 15	hbfv 2837	. . . . . . 7 ⊢ (u ∈ (G ‘y) → ∀z u ∈ (G ‘y))
23	16, 22	hbfv 2837	. . . . . 6 ⊢ (u ∈ (F ‘(G ‘y)) → ∀z u ∈ (F ‘(G ‘y)))
24		fveq2 2832	. . . . . 6 ⊢ (z = (G ‘y) → (F ‘z) = (F ‘(G ‘y)))
25	14, 15, 23, 7, 24	rdgsucopab 2984	. . . . 5 ⊢ ((y ∈ On ∧ (F ‘(G ‘y)) ∈ V) → (G ‘suc y) = (F ‘(G ‘y)))
26	13, 25	mpan2 519	. . . 4 ⊢ (y ∈ On → (G ‘suc y) = (F ‘(G ‘y)))
27		fveq2 2832	. . . . 5 ⊢ ((G ‘y) = x → (F ‘(G ‘y)) = (F ‘x))
28		id 9	. . . . 5 ⊢ ((F ‘x) = x → (F ‘x) = x)
29	27, 28	sylan9eqr 1145	. . . 4 ⊢ (((F ‘x) = x ∧ (G ‘y) = x) → (F ‘(G ‘y)) = x)
30	26, 29	sylan9eq 1144	. . 3 ⊢ ((y ∈ On ∧ ((F ‘x) = x ∧ (G ‘y) = x)) → (G ‘suc y) = x)
31	30	exp32 294	. 2 ⊢ (y ∈ On → ((F ‘x) = x → ((G ‘y) = x → (G ‘suc y) = x)))
32		visset 1350	. . . . . . . 8 ⊢ v ∈ V
33		rdglim2a 2988	. . . . . . . 8 ⊢ ((v ∈ V ∧ Lim v) → (rec({⟨z, w⟩∣w = (F ‘z)}, x) ‘v) = ∪y ∈ v (rec({⟨z, w⟩∣w = (F ‘z)}, x) ‘y))
34	32, 33	mpan 518	. . . . . . 7 ⊢ (Lim v → (rec({⟨z, w⟩∣w = (F ‘z)}, x) ‘v) = ∪y ∈ v (rec({⟨z, w⟩∣w = (F ‘z)}, x) ‘y))
35	7	fveq1i 2833	. . . . . . 7 ⊢ (G ‘v) = (rec({⟨z, w⟩∣w = (F ‘z)}, x) ‘v)
36	7	fveq1i 2833	. . . . . . . . 9 ⊢ (G ‘y) = (rec({⟨z, w⟩∣w = (F ‘z)}, x) ‘y)
37	36	a1i 7	. . . . . . . 8 ⊢ (y ∈ v → (G ‘y) = (rec({⟨z, w⟩∣w = (F ‘z)}, x) ‘y))
38	37	iuneq2i 2008	. . . . . . 7 ⊢ ∪y ∈ v (G ‘y) = ∪y ∈ v (rec({⟨z, w⟩∣w = (F ‘z)}, x) ‘y)
39	34, 35, 38	3eqtr4g 1147	. . . . . 6 ⊢ (Lim v → (G ‘v) = ∪y ∈ v (G ‘y))
40	39	adantr 306	. . . . 5 ⊢ ((Lim v ∧ ∀y ∈ v (G ‘y) = x) → (G ‘v) = ∪y ∈ v (G ‘y))
41		iuneq2 2006	. . . . . 6 ⊢ (∀y ∈ v (G ‘y) = x → ∪y ∈ v (G ‘y) = ∪y ∈ v x)
42		df-lim 2204	. . . . . . . 8 ⊢ (Lim v ↔ (Ord v ∧ ¬ v = ∅ ∧ v = ∪v))
43		3simp2 595	. . . . . . . 8 ⊢ ((Ord v ∧ ¬ v = ∅ ∧ v = ∪v) → ¬ v = ∅)
44	42, 43	sylbi 174	. . . . . . 7 ⊢ (Lim v → ¬ v = ∅)
45		iunconst 2000	. . . . . . 7 ⊢ (¬ v = ∅ → ∪y ∈ v x = x)
46	44, 45	syl 12	. . . . . 6 ⊢ (Lim v → ∪y ∈ v x = x)
47	41, 46	sylan9eqr 1145	. . . . 5 ⊢ ((Lim v ∧ ∀y ∈ v (G ‘y) = x) → ∪y ∈ v (G ‘y) = x)
48	40, 47	eqtrd 1128	. . . 4 ⊢ ((Lim v ∧ ∀y ∈ v (G ‘y) = x) → (G ‘v) = x)
49	48	exp 291	. . 3 ⊢ (Lim v → (∀y ∈ v (G ‘y) = x → (G ‘v) = x))
50	49	a1d 14	. 2 ⊢ (Lim v → ((F ‘x) = x → (∀y ∈ v (G ‘y) = x → (G ‘v) = x)))
51	2, 4, 6, 12, 31, 50	tfinds2 2405	1 ⊢ (v ∈ On → ((F ‘x) = x → (G ‘v) = x))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∧ w3a 581  ∀wal 672   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348  ∅c0 1707  ∪cuni 1919  ∪ciun 1994  {copab 2055  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201   ‘cfv 2422  reccrdg 2969

This theorem is referenced by:  abianfp 3000

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-fv 2438  df-rdg 2970

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