Theorem tfrlem9 5127

Description: Lemma for transfinite recursion. Here we compute the value of F (the union of all acceptable functions).

Hypotheses

Ref	Expression
tfrlem.1	|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
tfrlem.2	|- F = U.A

Assertion

Ref	Expression
tfrlem9	|- (y e. dom F -> (F` y) = (G` (F |` y)))

Distinct variable groups:   x,y,f,A   x,F,y,f   x,G,y,f

Proof of Theorem tfrlem9

Step	Hyp	Ref	Expression
1		visset 2295	. . 3 |- y e. _V
2	1	eldm2 4154	. 2 |- (y e. dom F <-> E.z<.y, z>. e. F)
3		tfrlem.2	. . . . . . 7 |- F = U.A
4		tfrlem.1	. . . . . . . 8 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
5	4	unieqi 3187	. . . . . . 7 |- U.A = U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
6	3, 5	eqtri 1908	. . . . . 6 |- F = U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
7	6	eleq2i 1961	. . . . 5 |- (<.y, z>. e. F <-> <.y, z>. e. U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))})
8		eluniab 3189	. . . . 5 |- (<.y, z>. e. U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))} <-> E.f(<.y, z>. e. f /\ E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))))
9	7, 8	bitri 190	. . . 4 |- (<.y, z>. e. F <-> E.f(<.y, z>. e. f /\ E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))))
10		fnop 4516	. . . . . . . . . . . . . 14 |- ((f Fn x /\ <.y, z>. e. f) -> y e. x)
11		ra4e 2156	. . . . . . . . . . . . . . . 16 |- ((x e. On /\ (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))) -> E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y))))
12	4	abeq2i 2001	. . . . . . . . . . . . . . . . 17 |- (f e. A <-> E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y))))
13		elssuni 3206	. . . . . . . . . . . . . . . . . 18 |- (f e. A -> f C_ U.A)
14	13, 3	syl6ssr 2664	. . . . . . . . . . . . . . . . 17 |- (f e. A -> f C_ F)
15	12, 14	sylbir 218	. . . . . . . . . . . . . . . 16 |- (E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y))) -> f C_ F)
16	11, 15	syl 12	. . . . . . . . . . . . . . 15 |- ((x e. On /\ (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))) -> f C_ F)
17		ra4 2155	. . . . . . . . . . . . . . . . . . 19 |- (A.y e. x (f` y) = (G` (f |` y)) -> (y e. x -> (f` y) = (G` (f |` y))))
18	17	com12 14	. . . . . . . . . . . . . . . . . 18 |- (y e. x -> (A.y e. x (f` y) = (G` (f |` y)) -> (f` y) = (G` (f |` y))))
19		fndm 4512	. . . . . . . . . . . . . . . . . . . . 21 |- (f Fn x -> dom f = x)
20	19	eleq2d 1964	. . . . . . . . . . . . . . . . . . . 20 |- (f Fn x -> (y e. dom f <-> y e. x))
21	4, 3	tfrlem7 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- Fun F
22		funssfv 4692	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((Fun F /\ f C_ F /\ y e. dom f) -> (F` y) = (f` y))
23	21, 22	mp3an1 1178	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((f C_ F /\ y e. dom f) -> (F` y) = (f` y))
24	23	adantrl 430	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((f C_ F /\ ((f Fn x /\ x e. On) /\ y e. dom f)) -> (F` y) = (f` y))
25		fun2ssres 4461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((Fun F /\ f C_ F /\ y C_ dom f) -> (F |` y) = (f |` y))
26	25	fveq2d 4685	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((Fun F /\ f C_ F /\ y C_ dom f) -> (G` (F |` y)) = (G` (f |` y)))
27	21, 26	mp3an1 1178	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((f C_ F /\ y C_ dom f) -> (G` (F |` y)) = (G` (f |` y)))
28	19	eleq1d 1963	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (f Fn x -> (dom f e. On <-> x e. On))
29		onelss 3705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (dom f e. On -> (y e. dom f -> y C_ dom f))
30	28, 29	syl6bir 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (f Fn x -> (x e. On -> (y e. dom f -> y C_ dom f)))
31	30	imp31 389	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (((f Fn x /\ x e. On) /\ y e. dom f) -> y C_ dom f)
32	27, 31	sylan2 500	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((f C_ F /\ ((f Fn x /\ x e. On) /\ y e. dom f)) -> (G` (F |` y)) = (G` (f |` y)))
33	24, 32	eqeq12d 1899	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((f C_ F /\ ((f Fn x /\ x e. On) /\ y e. dom f)) -> ((F` y) = (G` (F |` y)) <-> (f` y) = (G` (f |` y))))
34	33	exbiri 421	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (f C_ F -> (((f Fn x /\ x e. On) /\ y e. dom f) -> ((f` y) = (G` (f |` y)) -> (F` y) = (G` (F |` y)))))
35	34	com3l 38	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((f Fn x /\ x e. On) /\ y e. dom f) -> ((f` y) = (G` (f |` y)) -> (f C_ F -> (F` y) = (G` (F |` y)))))
36	35	exp31 407	. . . . . . . . . . . . . . . . . . . . . 22 |- (f Fn x -> (x e. On -> (y e. dom f -> ((f` y) = (G` (f |` y)) -> (f C_ F -> (F` y) = (G` (F |` y)))))))
37	36	com34 40	. . . . . . . . . . . . . . . . . . . . 21 |- (f Fn x -> (x e. On -> ((f` y) = (G` (f |` y)) -> (y e. dom f -> (f C_ F -> (F` y) = (G` (F |` y)))))))
38	37	com24 41	. . . . . . . . . . . . . . . . . . . 20 |- (f Fn x -> (y e. dom f -> ((f` y) = (G` (f |` y)) -> (x e. On -> (f C_ F -> (F` y) = (G` (F |` y)))))))
39	20, 38	sylbird 222	. . . . . . . . . . . . . . . . . . 19 |- (f Fn x -> (y e. x -> ((f` y) = (G` (f |` y)) -> (x e. On -> (f C_ F -> (F` y) = (G` (F |` y)))))))
40	39	com3l 38	. . . . . . . . . . . . . . . . . 18 |- (y e. x -> ((f` y) = (G` (f |` y)) -> (f Fn x -> (x e. On -> (f C_ F -> (F` y) = (G` (F |` y)))))))
41	18, 40	syld 30	. . . . . . . . . . . . . . . . 17 |- (y e. x -> (A.y e. x (f` y) = (G` (f |` y)) -> (f Fn x -> (x e. On -> (f C_ F -> (F` y) = (G` (F |` y)))))))
42	41	com24 41	. . . . . . . . . . . . . . . 16 |- (y e. x -> (x e. On -> (f Fn x -> (A.y e. x (f` y) = (G` (f |` y)) -> (f C_ F -> (F` y) = (G` (F |` y)))))))
43	42	imp4d 394	. . . . . . . . . . . . . . 15 |- (y e. x -> ((x e. On /\ (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))) -> (f C_ F -> (F` y) = (G` (F |` y)))))
44	16, 43	mpdi 59	. . . . . . . . . . . . . 14 |- (y e. x -> ((x e. On /\ (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))) -> (F` y) = (G` (F |` y))))
45	10, 44	syl 12	. . . . . . . . . . . . 13 |- ((f Fn x /\ <.y, z>. e. f) -> ((x e. On /\ (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))) -> (F` y) = (G` (F |` y))))
46	45	exp4d 412	. . . . . . . . . . . 12 |- ((f Fn x /\ <.y, z>. e. f) -> (x e. On -> (f Fn x -> (A.y e. x (f` y) = (G` (f |` y)) -> (F` y) = (G` (F |` y))))))
47	46	ex 402	. . . . . . . . . . 11 |- (f Fn x -> (<.y, z>. e. f -> (x e. On -> (f Fn x -> (A.y e. x (f` y) = (G` (f |` y)) -> (F` y) = (G` (F |` y)))))))
48	47	com4r 45	. . . . . . . . . 10 |- (f Fn x -> (f Fn x -> (<.y, z>. e. f -> (x e. On -> (A.y e. x (f` y) = (G` (f |` y)) -> (F` y) = (G` (F |` y)))))))
49	48	pm2.43i 78	. . . . . . . . 9 |- (f Fn x -> (<.y, z>. e. f -> (x e. On -> (A.y e. x (f` y) = (G` (f |` y)) -> (F` y) = (G` (F |` y))))))
50	49	com3l 38	. . . . . . . 8 |- (<.y, z>. e. f -> (x e. On -> (f Fn x -> (A.y e. x (f` y) = (G` (f |` y)) -> (F` y) = (G` (F |` y))))))
51	50	imp4a 391	. . . . . . 7 |- (<.y, z>. e. f -> (x e. On -> ((f Fn x /\ A.y e. x (f` y) = (G` (f |` y))) -> (F` y) = (G` (F |` y)))))
52	51	r19.23adv 2215	. . . . . 6 |- (<.y, z>. e. f -> (E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y))) -> (F` y) = (G` (F |` y))))
53	52	imp 377	. . . . 5 |- ((<.y, z>. e. f /\ E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))) -> (F` y) = (G` (F |` y)))
54	53	19.23aiv 1674	. . . 4 |- (E.f(<.y, z>. e. f /\ E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))) -> (F` y) = (G` (F |` y)))
55	9, 54	sylbi 216	. . 3 |- (<.y, z>. e. F -> (F` y) = (G` (F |` y)))
56	55	19.23aiv 1674	. 2 |- (E.z<.y, z>. e. F -> (F` y) = (G` (F |` y)))
57	2, 56	sylbi 216	1 |- (y e. dom F -> (F` y) = (G` (F |` y)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106   C_ wss 2593  <.cop 3046  U.cuni 3177  Oncon0 3657  dom cdm 3986   |` cres 3988  Fun wfun 3992   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  tfrlem11 5129  tfr2 5133

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-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-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-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-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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