Theorem tz7.44-3 5138

Description: The value of F at a limit ordinal. Part 3 of Theorem 7.44 of [TakeutiZaring] p. 49.

Hypotheses

Ref	Expression
tz7.44.1	|- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}
tz7.44.2	|- F Fn On
tz7.44.3	|- (x e. On -> (F` x) = (G` (F |` x)))
tz7.44.5	|- B e. On

Assertion

Ref	Expression
tz7.44-3	|- (Lim B -> (F` B) = U.(F"B))

Distinct variable groups:   x,y,A   x,F   x,G   y,H   x,B,y   y,F   x,H

Proof of Theorem tz7.44-3

Step	Hyp	Ref	Expression
1		tz7.44.2	. . . . . . . . . 10 |- F Fn On
2		fndm 4512	. . . . . . . . . 10 |- (F Fn On -> dom F = On)
3	1, 2	ax-mp 7	. . . . . . . . 9 |- dom F = On
4	3	ineq2i 2793	. . . . . . . 8 |- (B i^i dom F) = (B i^i On)
5		dmres 4234	. . . . . . . 8 |- dom ( F |` B) = (B i^i dom F)
6		tz7.44.5	. . . . . . . . . 10 |- B e. On
7	6	onssi 3918	. . . . . . . . 9 |- B C_ On
8		dfss 2606	. . . . . . . . 9 |- (B C_ On <-> B = (B i^i On))
9	7, 8	mpbi 206	. . . . . . . 8 |- B = (B i^i On)
10	4, 5, 9	3eqtr4i 1921	. . . . . . 7 |- dom ( F |` B) = B
11		limeq 3669	. . . . . . 7 |- (dom ( F |` B) = B -> (Lim dom ( F |` B) <-> Lim B))
12	10, 11	ax-mp 7	. . . . . 6 |- (Lim dom ( F |` B) <-> Lim B)
13	12	biimpri 169	. . . . 5 |- (Lim B -> Lim dom ( F |` B))
14		df-ima 4007	. . . . . 6 |- (F"B) = ran ( F |` B)
15	14	unieqi 3187	. . . . 5 |- U.(F"B) = U.ran ( F |` B)
16	13, 15	jctir 317	. . . 4 |- (Lim B -> (Lim dom ( F |` B) /\ U.(F"B) = U.ran ( F |` B)))
17		fnfun 4510	. . . . . . 7 |- (F Fn On -> Fun F)
18	1, 17	ax-mp 7	. . . . . 6 |- Fun F
19		resfunexg 4500	. . . . . 6 |- ((Fun F /\ B e. On) -> (F |` B) e. _V)
20	18, 6, 19	mp2an 761	. . . . 5 |- (F |` B) e. _V
21	6	elisseti 2301	. . . . . . . 8 |- B e. _V
22	21	funimaex 4496	. . . . . . 7 |- (Fun F -> (F"B) e. _V)
23	18, 22	ax-mp 7	. . . . . 6 |- (F"B) e. _V
24	23	uniex 3794	. . . . 5 |- U.(F"B) e. _V
25		dmeq 4157	. . . . . . 7 |- (x = (F |` B) -> dom x = dom ( F |` B))
26		limeq 3669	. . . . . . 7 |- (dom x = dom ( F |` B) -> (Lim dom x <-> Lim dom ( F |` B)))
27	25, 26	syl 12	. . . . . 6 |- (x = (F |` B) -> (Lim dom x <-> Lim dom ( F |` B)))
28		rneq 4186	. . . . . . . 8 |- (x = (F |` B) -> ran x = ran ( F |` B))
29	28	unieqd 3188	. . . . . . 7 |- (x = (F |` B) -> U.ran x = U.ran ( F |` B))
30	29	eqeq2d 1895	. . . . . 6 |- (x = (F |` B) -> (y = U.ran x <-> y = U.ran ( F |` B)))
31	27, 30	anbi12d 690	. . . . 5 |- (x = (F |` B) -> ((Lim dom x /\ y = U.ran x) <-> (Lim dom ( F |` B) /\ y = U.ran ( F |` B))))
32		eqeq1 1890	. . . . . 6 |- (y = U.(F"B) -> (y = U.ran ( F |` B) <-> U.(F"B) = U.ran ( F |` B)))
33	32	anbi2d 678	. . . . 5 |- (y = U.(F"B) -> ((Lim dom ( F |` B) /\ y = U.ran ( F |` B)) <-> (Lim dom ( F |` B) /\ U.(F"B) = U.ran ( F |` B))))
34	20, 24, 31, 33	opelopab 3570	. . . 4 |- (<.(F |` B), U.(F"B)>. e. {<.x, y>. | (Lim dom x /\ y = U.ran x)} <-> (Lim dom ( F |` B) /\ U.(F"B) = U.ran ( F |` B)))
35	16, 34	sylibr 217	. . 3 |- (Lim B -> <.(F |` B), U.(F"B)>. e. {<.x, y>. | (Lim dom x /\ y = U.ran x)})
36		3mix3 1047	. . . . . 6 |- ((Lim dom x /\ y = U.ran x) -> ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x)))
37	36	ssopab2i 3574	. . . . 5 |- {<.x, y>. | (Lim dom x /\ y = U.ran x)} C_ {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}
38		tz7.44.1	. . . . 5 |- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}
39	37, 38	sseqtr4i 2650	. . . 4 |- {<.x, y>. | (Lim dom x /\ y = U.ran x)} C_ G
40	39	sseli 2617	. . 3 |- (<.(F |` B), U.(F"B)>. e. {<.x, y>. | (Lim dom x /\ y = U.ran x)} -> <.(F |` B), U.(F"B)>. e. G)
41	38	tz7.44lem1 5135	. . . 4 |- Fun G
42	24	funopfv 4710	. . . 4 |- (Fun G -> (<.(F |` B), U.(F"B)>. e. G -> (G` (F |` B)) = U.(F"B)))
43	41, 42	ax-mp 7	. . 3 |- (<.(F |` B), U.(F"B)>. e. G -> (G` (F |` B)) = U.(F"B))
44	35, 40, 43	3syl 24	. 2 |- (Lim B -> (G` (F |` B)) = U.(F"B))
45		fveq2 4681	. . . . 5 |- (x = B -> (F` x) = (F` B))
46		reseq2 4219	. . . . . 6 |- (x = B -> (F |` x) = (F |` B))
47	46	fveq2d 4685	. . . . 5 |- (x = B -> (G` (F |` x)) = (G` (F |` B)))
48	45, 47	eqeq12d 1899	. . . 4 |- (x = B -> ((F` x) = (G` (F |` x)) <-> (F` B) = (G` (F |` B))))
49		tz7.44.3	. . . 4 |- (x e. On -> (F` x) = (G` (F |` x)))
50	48, 49	vtoclga 2352	. . 3 |- (B e. On -> (F` B) = (G` (F |` B)))
51	6, 50	ax-mp 7	. 2 |- (F` B) = (G` (F |` B))
52	44, 51	syl5eq 1940	1 |- (Lim B -> (F` B) = U.(F"B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  <.cop 3046  U.cuni 3177  {copab 3395  Oncon0 3657  Lim wlim 3658  dom cdm 3986  ran crn 3987   |` cres 3988  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  rdglimi 5151

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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  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-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

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