Theorem abianfplem 3946

Description: Lemma for abianfp 3947. 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 3709	. . 3 |- (v = (/) -> (G` v) = (G` (/)))
2	1	eqeq1d 1475	. 2 |- (v = (/) -> ((G` v) = x <-> (G` (/)) = x))
3		fveq2 3709	. . 3 |- (v = y -> (G` v) = (G` y))
4	3	eqeq1d 1475	. 2 |- (v = y -> ((G` v) = x <-> (G` y) = x))
5		fveq2 3709	. . 3 |- (v = suc y -> (G` v) = (G` suc y))
6	5	eqeq1d 1475	. 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 3710	. . . 4 |- (G` (/)) = (rec({<.z, w>. | w = (F` z)}, x)` (/))
9		visset 1804	. . . . 5 |- x e. V
10	9	rdg0 3926	. . . 4 |- (rec({<.z, w>. | w = (F` z)}, x)` (/)) = x
11	8, 10	eqtr 1487	. . 3 |- (G` (/)) = x
12	11	a1i 8	. 2 |- ((F` x) = x -> (G` (/)) = x)
13		fvex 3717	. . . . 5 |- (F` (G` y)) e. V
14		ax-17 968	. . . . . 6 |- (u e. x -> A.z u e. x)
15		ax-17 968	. . . . . 6 |- (u e. y -> A.z u e. y)
16		ax-17 968	. . . . . . 7 |- (u e. F -> A.z u e. F)
17		hbopab1 2802	. . . . . . . . . 10 |- (u e. {<.z, w>. | w = (F` z)} -> A.z u e. {<.z, w>. | w = (F` z)})
18	17, 14	hbrdg 3921	. . . . . . . . 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 1555	. . . . . . . 8 |- (u e. G -> A.z u e. G)
20	19, 15	hbfv 3714	. . . . . . 7 |- (u e. (G` y) -> A.z u e. (G` y))
21	16, 20	hbfv 3714	. . . . . 6 |- (u e. (F` (G` y)) -> A.z u e. (F` (G` y)))
22		fveq2 3709	. . . . . 6 |- (z = (G` y) -> (F` z) = (F` (G` y)))
23	14, 15, 21, 7, 22	rdgsucopab 3931	. . . . 5 |- ((y e. On /\ (F` (G` y)) e. V) -> (G` suc y) = (F` (G` y)))
24	13, 23	mpan2 694	. . . 4 |- (y e. On -> (G` suc y) = (F` (G` y)))
25		fveq2 3709	. . . . 5 |- ((G` y) = x -> (F` (G` y)) = (F` x))
26		id 59	. . . . 5 |- ((F` x) = x -> (F` x) = x)
27	25, 26	sylan9eqr 1521	. . . 4 |- (((F` x) = x /\ (G` y) = x) -> (F` (G` y)) = x)
28	24, 27	sylan9eq 1519	. . 3 |- ((y e. On /\ ((F` x) = x /\ (G` y) = x)) -> (G` suc y) = x)
29	28	exp32 377	. 2 |- (y e. On -> ((F` x) = x -> ((G` y) = x -> (G` suc y) = x)))
30		visset 1804	. . . . . . . 8 |- v e. V
31		rdglim2a 3935	. . . . . . . 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 693	. . . . . . 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 3710	. . . . . . 7 |- (G` v) = (rec({<.z, w>. | w = (F` z)}, x)` v)
34	7	fveq1i 3710	. . . . . . . . 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 2570	. . . . . . 7 |- U_y e. v (G` y) = U_y e. v (rec({<.z, w>. | w = (F` z)}, x)` y)
37	32, 33, 36	3eqtr4g 1523	. . . . . 6 |- (Lim v -> (G` v) = U_y e. v (G` y))
38	37	adantr 389	. . . . 5 |- ((Lim v /\ A.y e. v (G` y) = x) -> (G` v) = U_y e. v (G` y))
39		iuneq2 2568	. . . . . 6 |- (A.y e. v (G` y) = x -> U_y e. v (G` y) = U_y e. v x)
40		df-lim 2943	. . . . . . . 8 |- (Lim v <-> (Ord v /\ v =/= (/) /\ v = U.v))
41		3simp2 787	. . . . . . . 8 |- ((Ord v /\ v =/= (/) /\ v = U.v) -> v =/= (/))
42	40, 41	sylbi 199	. . . . . . 7 |- (Lim v -> v =/= (/))
43		iunconst 2562	. . . . . . 7 |- (v =/= (/) -> U_y e. v x = x)
44	42, 43	syl 10	. . . . . 6 |- (Lim v -> U_y e. v x = x)
45	39, 44	sylan9eqr 1521	. . . . 5 |- ((Lim v /\ A.y e. v (G` y) = x) -> U_y e. v (G` y) = x)
46	38, 45	eqtrd 1499	. . . 4 |- ((Lim v /\ A.y e. v (G` y) = x) -> (G` v) = x)
47	46	ex 373	. . 3 |- (Lim v -> (A.y e. v (G` y) = x -> (G` v) = x))
48	47	a1d 12	. 2 |- (Lim v -> ((F` x) = x -> (A.y e. v (G` y) = x -> (G` v) = x)))
49	2, 4, 6, 12, 29, 48	tfinds2 3155	1 |- (v e. On -> ((F` x) = x -> (G` v) = x))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   =/= wne 1577  A.wral 1637  Vcvv 1802  (/)c0 2270  U.cuni 2493  U_ciun 2556  {copab 2656  Ord word 2937  Oncon0 2938  Lim wlim 2939  suc csuc 2940  ` cfv 3172  reccrdg 3916

This theorem is referenced by:  abianfp 3947

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917

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