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 e. V
abianfp.2	|- G = rec({<.z, w>. | w = (F` z)}, x)

Assertion

Ref	Expression
abianfplem	|- (v e. 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 e. 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)) e. V
14		ax-17 925	. . . . . 6 |- (u e. x -> A.z u e. x)
15		ax-17 925	. . . . . 6 |- (u e. y -> A.z u e. y)
16		ax-17 925	. . . . . . 7 |- (u e. F -> A.z u e. F)
17		hbopab1 2112	. . . . . . . . . 10 |- (u e. {<.z, w>. | w = (F` z)} -> A.z u e. {<.z, w>. | w = (F` z)})
18	17, 14	hbrdg 2974	. . . . . . . . 9 |- (u e. rec({<.z, w>. | w = (F` z)}, x) -> A.z u e. rec({<.z, w>. | w = (F` z)}, x))
19	7	eleq2i 1153	. . . . . . . . 9 |- (u e. G <-> u e. rec({<.z, w>. | w = (F` z)}, x))
20	19	bial 695	. . . . . . . . 9 |- (A.z u e. G <-> A.z u e. rec({<.z, w>. | w = (F` z)}, x))
21	18, 19, 20	3imtr4 192	. . . . . . . 8 |- (u e. G -> A.z u e. G)
22	21, 15	hbfv 2837	. . . . . . 7 |- (u e. (G` y) -> A.z u e. (G` y))
23	16, 22	hbfv 2837	. . . . . 6 |- (u e. (F` (G` y)) -> A.z u e. (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 e. On /\ (F` (G` y)) e. V) -> (G` suc y) = (F` (G` y)))
26	13, 25	mpan2 519	. . . 4 |- (y e. 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 e. On /\ ((F` x) = x /\ (G` y) = x)) -> (G` suc y) = x)
31	30	exp32 294	. 2 |- (y e. On -> ((F` x) = x -> ((G` y) = x -> (G` suc y) = x)))
32		visset 1350	. . . . . . . 8 |- v e. V
33		rdglim2a 2988	. . . . . . . 8 |- ((v e. V /\ Lim v) -> (rec({<.z, w>. | w = (F` z)}, x)` v) = U.y e. 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) = U.y e. 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 e. v -> (G` y) = (rec({<.z, w>. | w = (F` z)}, x)` y))
38	37	iuneq2i 2008	. . . . . . 7 |- U.y e. v (G` y) = U.y e. v (rec({<.z, w>. | w = (F` z)}, x)` y)
39	34, 35, 38	3eqtr4g 1147	. . . . . 6 |- (Lim v -> (G` v) = U.y e. v (G` y))
40	39	adantr 306	. . . . 5 |- ((Lim v /\ A.y e. v (G` y) = x) -> (G` v) = U.y e. v (G` y))
41		iuneq2 2006	. . . . . 6 |- (A.y e. v (G` y) = x -> U.y e. v (G` y) = U.y e. v x)
42		df-lim 2204	. . . . . . . 8 |- (Lim v <-> (Ord v /\ -. v = (/) /\ v = U.v))
43		3simp2 595	. . . . . . . 8 |- ((Ord v /\ -. v = (/) /\ v = U.v) -> -. v = (/))
44	42, 43	sylbi 174	. . . . . . 7 |- (Lim v -> -. v = (/))
45		iunconst 2000	. . . . . . 7 |- (-. v = (/) -> U.y e. v x = x)
46	44, 45	syl 12	. . . . . 6 |- (Lim v -> U.y e. v x = x)
47	41, 46	sylan9eqr 1145	. . . . 5 |- ((Lim v /\ A.y e. v (G` y) = x) -> U.y e. v (G` y) = x)
48	40, 47	eqtrd 1128	. . . 4 |- ((Lim v /\ A.y e. v (G` y) = x) -> (G` v) = x)
49	48	exp 291	. . 3 |- (Lim v -> (A.y e. v (G` y) = x -> (G` v) = x))
50	49	a1d 14	. 2 |- (Lim v -> ((F` x) = x -> (A.y e. v (G` y) = x -> (G` v) = x)))
51	2, 4, 6, 12, 31, 50	tfinds2 2405	1 |- (v e. On -> ((F` x) = x -> (G` v) = x))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   /\ w3a 581  A.wal 672   = weq 797   e. wel 803   = wceq 1091   e. wcel 1092  A.wral 1201  Vcvv 1348  (/)c0 1707  U.cuni 1919  U.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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