Theorem abianfplem 5170

Description: Lemma for abianfp 5171. 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 groups:   x,v,A   x,z,w,F,v   v,G

Proof of Theorem abianfplem

Step	Hyp	Ref	Expression
1		fveq2 4681	. . 3 |- (v = (/) -> (G` v) = (G` (/)))
2	1	eqeq1d 1892	. 2 |- (v = (/) -> ((G` v) = x <-> (G` (/)) = x))
3		fveq2 4681	. . 3 |- (v = y -> (G` v) = (G` y))
4	3	eqeq1d 1892	. 2 |- (v = y -> ((G` v) = x <-> (G` y) = x))
5		fveq2 4681	. . 3 |- (v = suc y -> (G` v) = (G` suc y))
6	5	eqeq1d 1892	. 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 4682	. . . 4 |- (G` (/)) = (rec({<.z, w>. | w = (F` z)}, x)` (/))
9		visset 2295	. . . . 5 |- x e. _V
10	9	rdg0 5149	. . . 4 |- (rec({<.z, w>. | w = (F` z)}, x)` (/)) = x
11	8, 10	eqtri 1908	. . 3 |- (G` (/)) = x
12	11	a1i 8	. 2 |- ((F` x) = x -> (G` (/)) = x)
13		fvex 4689	. . . . 5 |- (F` (G` y)) e. _V
14		ax-17 1317	. . . . . 6 |- (u e. x -> A.z u e. x)
15		ax-17 1317	. . . . . 6 |- (u e. y -> A.z u e. y)
16		ax-17 1317	. . . . . . 7 |- (u e. F -> A.z u e. F)
17		hbopab1 3562	. . . . . . . . . 10 |- (u e. {<.z, w>. | w = (F` z)} -> A.z u e. {<.z, w>. | w = (F` z)})
18	17, 14	hbrdg 5144	. . . . . . . . 9 |- (u e. rec({<.z, w>. | w = (F` z)}, x) -> A.z u e. rec({<.z, w>. | w = (F` z)}, x))
19	7, 18	hbxfr 1992	. . . . . . . 8 |- (u e. G -> A.z u e. G)
20	19, 15	hbfv 4686	. . . . . . 7 |- (u e. (G` y) -> A.z u e. (G` y))
21	16, 20	hbfv 4686	. . . . . 6 |- (u e. (F` (G` y)) -> A.z u e. (F` (G` y)))
22		fveq2 4681	. . . . . 6 |- (z = (G` y) -> (F` z) = (F` (G` y)))
23	14, 15, 21, 7, 22	rdgsucopab 5154	. . . . 5 |- ((y e. On /\ (F` (G` y)) e. _V) -> (G` suc y) = (F` (G` y)))
24	13, 23	mpan2 760	. . . 4 |- (y e. On -> (G` suc y) = (F` (G` y)))
25		fveq2 4681	. . . . 5 |- ((G` y) = x -> (F` (G` y)) = (F` x))
26		id 73	. . . . 5 |- ((F` x) = x -> (F` x) = x)
27	25, 26	sylan9eqr 1951	. . . 4 |- (((F` x) = x /\ (G` y) = x) -> (F` (G` y)) = x)
28	24, 27	sylan9eq 1948	. . 3 |- ((y e. On /\ ((F` x) = x /\ (G` y) = x)) -> (G` suc y) = x)
29	28	exp32 408	. 2 |- (y e. On -> ((F` x) = x -> ((G` y) = x -> (G` suc y) = x)))
30		visset 2295	. . . . . . . 8 |- v e. _V
31		rdglim2a 5158	. . . . . . . 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))
32	30, 31	mpan 759	. . . . . . 7 |- (Lim v -> (rec({<.z, w>. | w = (F` z)}, x)` v) = U_y e. v (rec({<.z, w>. | w = (F` z)}, x)` y))
33	7	fveq1i 4682	. . . . . . 7 |- (G` v) = (rec({<.z, w>. | w = (F` z)}, x)` v)
34	7	fveq1i 4682	. . . . . . . . 9 |- (G` y) = (rec({<.z, w>. | w = (F` z)}, x)` y)
35	34	a1i 8	. . . . . . . 8 |- (y e. v -> (G` y) = (rec({<.z, w>. | w = (F` z)}, x)` y))
36	35	iuneq2i 3276	. . . . . . 7 |- U_y e. v (G` y) = U_y e. v (rec({<.z, w>. | w = (F` z)}, x)` y)
37	32, 33, 36	3eqtr4g 1953	. . . . . 6 |- (Lim v -> (G` v) = U_y e. v (G` y))
38	37	adantr 425	. . . . 5 |- ((Lim v /\ A.y e. v (G` y) = x) -> (G` v) = U_y e. v (G` y))
39		iuneq2 3273	. . . . . 6 |- (A.y e. v (G` y) = x -> U_y e. v (G` y) = U_y e. v x)
40		df-lim 3662	. . . . . . . 8 |- (Lim v <-> (Ord v /\ v =/= (/) /\ v = U.v))
41	40	simp2bi 892	. . . . . . 7 |- (Lim v -> v =/= (/))
42		iunconst 3262	. . . . . . 7 |- (v =/= (/) -> U_y e. v x = x)
43	41, 42	syl 12	. . . . . 6 |- (Lim v -> U_y e. v x = x)
44	39, 43	sylan9eqr 1951	. . . . 5 |- ((Lim v /\ A.y e. v (G` y) = x) -> U_y e. v (G` y) = x)
45	38, 44	eqtrd 1925	. . . 4 |- ((Lim v /\ A.y e. v (G` y) = x) -> (G` v) = x)
46	45	ex 402	. . 3 |- (Lim v -> (A.y e. v (G` y) = x -> (G` v) = x))
47	46	a1d 15	. 2 |- (Lim v -> ((F` x) = x -> (A.y e. v (G` y) = x -> (G` v) = x)))
48	2, 4, 6, 12, 29, 47	tfinds2 3947	1 |- (v e. On -> ((F` x) = x -> (G` v) = x))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  _Vcvv 2292  (/)c0 2875  U.cuni 3177  U_ciun 3255  {copab 3395  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659  ` cfv 3998  reccrdg 5139

This theorem is referenced by:  abianfp 5171

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-fv 4014  df-rdg 5140

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