Theorem tz7.44-3 2968

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 group(s):   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 2723	. . . . . . . . . 10 |- (F Fn On -> dom F = On)
3	1, 2	ax-mp 6	. . . . . . . . 9 |- dom F = On
4	3	ineq2i 1642	. . . . . . . 8 |- (B i^i dom F) = (B i^i On)
5		dmres 2584	. . . . . . . 8 |- dom (F |` B) = (B i^i dom F)
6		tz7.44.5	. . . . . . . . . 10 |- B e. On
7	6	onss 2347	. . . . . . . . 9 |- B (_ On
8		dfss 1493	. . . . . . . . 9 |- (B (_ On <-> B = (B i^i On))
9	7, 8	mpbi 164	. . . . . . . 8 |- B = (B i^i On)
10	4, 5, 9	3eqtr4 1126	. . . . . . 7 |- dom (F |` B) = B
11		limeq 2211	. . . . . . 7 |- (dom (F |` B) = B -> (Lim dom (F |` B) <-> Lim B))
12	10, 11	ax-mp 6	. . . . . 6 |- (Lim dom (F |` B) <-> Lim B)
13	12	biimpr 134	. . . . 5 |- (Lim B -> Lim dom (F |` B))
14		df-ima 2431	. . . . . 6 |- (F"B) = ran (F |` B)
15	14	unieqi 1928	. . . . 5 |- U.(F"B) = U.ran (F |` B)
16	13, 15	jctir 241	. . . 4 |- (Lim B -> (Lim dom (F |` B) /\ U.(F"B) = U.ran (F |` B)))
17		fnfun 2721	. . . . . . 7 |- (F Fn On -> Fun F)
18	1, 17	ax-mp 6	. . . . . 6 |- Fun F
19		resfunexg 2717	. . . . . 6 |- (B e. On -> (Fun F -> (F |` B) e. V))
20	6, 18, 19	mp2 43	. . . . 5 |- (F |` B) e. V
21	6	elisseti 1355	. . . . . . . 8 |- B e. V
22	21	funimaex 2716	. . . . . . 7 |- (Fun F -> (F"B) e. V)
23	18, 22	ax-mp 6	. . . . . 6 |- (F"B) e. V
24	23	uniex 1947	. . . . 5 |- U.(F"B) e. V
25		dmeq 2531	. . . . . . 7 |- (x = (F |` B) -> dom x = dom (F |` B))
26		limeq 2211	. . . . . . 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 2555	. . . . . . . 8 |- (x = (F |` B) -> ran x = ran (F |` B))
29	28	unieqd 1929	. . . . . . 7 |- (x = (F |` B) -> U.ran x = U.ran (F |` B))
30	29	cleq2d 1112	. . . . . 6 |- (x = (F |` B) -> (y = U.ran x <-> y = U.ran (F |` B)))
31	27, 30	anbi12d 476	. . . . 5 |- (x = (F |` B) -> ((Lim dom x /\ y = U.ran x) <-> (Lim dom (F |` B) /\ y = U.ran (F |` B))))
32		cleq1 1107	. . . . . 6 |- (y = U.(F"B) -> (y = U.ran (F |` B) <-> U.(F"B) = U.ran (F |` B)))
33	32	anbi2d 468	. . . . 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 2117	. . . 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 175	. . 3 |- (Lim B -> <.(F |` B), U.(F"B)>. e. {<.x, y>. | (Lim dom x /\ y = U.ran x)})
36		3mix3 602	. . . . . 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 2120	. . . . 5 |- {<.x, y>. | (Lim dom x /\ y = U.ran x)} (_ {<.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	sseqtr4 1533	. . . 4 |- {<.x, y>. | (Lim dom x /\ y = U.ran x)} (_ G
40	39	sseli 1504	. . 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 2965	. . . 4 |- Fun G
42	24	funfvopi 2853	. . . 4 |- (Fun G -> (<.(F |` B), U.(F"B)>. e. G -> (G` (F |` B)) = U.(F"B)))
43	41, 42	ax-mp 6	. . 3 |- (<.(F |` B), U.(F"B)>. e. G -> (G` (F |` B)) = U.(F"B))
44	35, 40, 43	3syl 21	. 2 |- (Lim B -> (G` (F |` B)) = U.(F"B))
45		fveq2 2832	. . . . 5 |- (x = B -> (F` x) = (F` B))
46		reseq2 2576	. . . . . 6 |- (x = B -> (F |` x) = (F |` B))
47	46	fveq2d 2836	. . . . 5 |- (x = B -> (G` (F |` x)) = (G` (F |` B)))
48	45, 47	cleq12d 1115	. . . 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 1387	. . 3 |- (B e. On -> (F` B) = (G` (F |` B)))
51	6, 50	ax-mp 6	. 2 |- (F` B) = (G` (F |` B))
52	44, 51	syl5eq 1136	1 |- (Lim B -> (F` B) = U.(F"B))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   \/ w3o 580   = wceq 1091   e. wcel 1092  Vcvv 1348   i^i cin 1486   (_ wss 1487  (/)c0 1707  <.cop 1810  U.cuni 1919  {copab 2055  Oncon0 2199  Lim wlim 2200  dom cdm 2410  ran crn 2411   |` cres 2412  "cima 2413  Fun wfun 2416   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  rdglim 2981

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-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  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-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

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