Theorem tz7.44-3 3921

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 3579	. . . . . . . . . 10 |- (F Fn On -> dom F = On)
3	1, 2	ax-mp 7	. . . . . . . . 9 |- dom F = On
4	3	ineq2i 2210	. . . . . . . 8 |- (B i^i dom F) = (B i^i On)
5		dmres 3372	. . . . . . . 8 |- dom ( F |` B) = (B i^i dom F)
6		tz7.44.5	. . . . . . . . . 10 |- B e. On
7	6	onss 3094	. . . . . . . . 9 |- B (_ On
8		dfss 2050	. . . . . . . . 9 |- (B (_ On <-> B = (B i^i On))
9	7, 8	mpbi 189	. . . . . . . 8 |- B = (B i^i On)
10	4, 5, 9	3eqtr4 1502	. . . . . . 7 |- dom ( F |` B) = B
11		limeq 2955	. . . . . . 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	biimpr 152	. . . . 5 |- (Lim B -> Lim dom ( F |` B))
14		df-ima 3186	. . . . . 6 |- (F"B) = ran ( F |` B)
15	14	unieqi 2506	. . . . 5 |- U.(F"B) = U.ran ( F |` B)
16	13, 15	jctir 293	. . . 4 |- (Lim B -> (Lim dom ( F |` B) /\ U.(F"B) = U.ran ( F |` B)))
17		fnfun 3577	. . . . . . 7 |- (F Fn On -> Fun F)
18	1, 17	ax-mp 7	. . . . . 6 |- Fun F
19		resfunexg 3571	. . . . . 6 |- ((Fun F /\ B e. On) -> (F |` B) e. V)
20	18, 6, 19	mp2an 696	. . . . 5 |- (F |` B) e. V
21	6	elisseti 1814	. . . . . . . 8 |- B e. V
22	21	funimaex 3568	. . . . . . 7 |- (Fun F -> (F"B) e. V)
23	18, 22	ax-mp 7	. . . . . 6 |- (F"B) e. V
24	23	uniex 2865	. . . . 5 |- U.(F"B) e. V
25		dmeq 3306	. . . . . . 7 |- (x = (F |` B) -> dom x = dom ( F |` B))
26		limeq 2955	. . . . . . 7 |- (dom x = dom ( F |` B) -> (Lim dom x <-> Lim dom ( F |` B)))
27	25, 26	syl 10	. . . . . 6 |- (x = (F |` B) -> (Lim dom x <-> Lim dom ( F |` B)))
28		rneq 3334	. . . . . . . 8 |- (x = (F |` B) -> ran x = ran ( F |` B))
29	28	unieqd 2507	. . . . . . 7 |- (x = (F |` B) -> U.ran x = U.ran ( F |` B))
30	29	eqeq2d 1483	. . . . . 6 |- (x = (F |` B) -> (y = U.ran x <-> y = U.ran ( F |` B)))
31	27, 30	anbi12d 627	. . . . 5 |- (x = (F |` B) -> ((Lim dom x /\ y = U.ran x) <-> (Lim dom ( F |` B) /\ y = U.ran ( F |` B))))
32		eqeq1 1478	. . . . . 6 |- (y = U.(F"B) -> (y = U.ran ( F |` B) <-> U.(F"B) = U.ran ( F |` B)))
33	32	anbi2d 615	. . . . 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 2815	. . . 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 200	. . 3 |- (Lim B -> <.(F |` B), U.(F"B)>. e. {<.x, y>. | (Lim dom x /\ y = U.ran x)})
36		3mix3 816	. . . . . 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 2818	. . . . 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 2090	. . . 4 |- {<.x, y>. | (Lim dom x /\ y = U.ran x)} (_ G
40	39	sseli 2061	. . 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 3918	. . . 4 |- Fun G
42	24	funopfv 3742	. . . 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 20	. 2 |- (Lim B -> (G` (F |` B)) = U.(F"B))
45		fveq2 3715	. . . . 5 |- (x = B -> (F` x) = (F` B))
46		reseq2 3361	. . . . . 6 |- (x = B -> (F |` x) = (F |` B))
47	46	fveq2d 3719	. . . . 5 |- (x = B -> (G` (F |` x)) = (G` (F |` B)))
48	45, 47	eqeq12d 1486	. . . 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 1848	. . 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 1516	1 |- (Lim B -> (F` B) = U.(F"B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 773   = wceq 954   e. wcel 956  Vcvv 1807   i^i cin 2042   (_ wss 2043  (/)c0 2276  <.cop 2407  U.cuni 2498  {copab 2661  Oncon0 2943  Lim wlim 2944  dom cdm 3165  ran crn 3166   |` cres 3167  "cima 3168  Fun wfun 3171   Fn wfn 3172  ` cfv 3177

This theorem is referenced by:  rdglim 3934

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193

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