Theorem tfr3 5134

Description: Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47. Finally we show that F is unique. We do this by showing that any class B with the same properties of F that we showed in parts 1 and 2 is identical to F.

Hypotheses

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

Assertion

Ref	Expression
tfr3	|- ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> B = F)

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

Proof of Theorem tfr3

Step	Hyp	Ref	Expression
1		ax-17 1317	. . . 4 |- (B Fn On -> A.x B Fn On)
2		hbra1 2147	. . . 4 |- (A.x e. On (B` x) = (G` (B |` x)) -> A.xA.x e. On (B` x) = (G` (B |` x)))
3	1, 2	hban 1356	. . 3 |- ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> A.x(B Fn On /\ A.x e. On (B` x) = (G` (B |` x))))
4		ax-17 1317	. . . . . 6 |- ((B` y) = (F` y) -> A.x(B` y) = (F` y))
5	3, 4	hbim 1354	. . . . 5 |- (((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (B` y) = (F` y)) -> A.x((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (B` y) = (F` y)))
6		fveq2 4681	. . . . . . 7 |- (x = y -> (B` x) = (B` y))
7		fveq2 4681	. . . . . . 7 |- (x = y -> (F` x) = (F` y))
8	6, 7	eqeq12d 1899	. . . . . 6 |- (x = y -> ((B` x) = (F` x) <-> (B` y) = (F` y)))
9	8	imbi2d 674	. . . . 5 |- (x = y -> (((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (B` x) = (F` x)) <-> ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (B` y) = (F` y))))
10		ra4 2155	. . . . . . . . . 10 |- (A.x e. On (B` x) = (G` (B |` x)) -> (x e. On -> (B` x) = (G` (B |` x))))
11		tfr.1	. . . . . . . . . . . . . . . . . . . . . 22 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
12		tfr.2	. . . . . . . . . . . . . . . . . . . . . 22 |- F = U.A
13	11, 12	tfr1 5132	. . . . . . . . . . . . . . . . . . . . 21 |- F Fn On
14		fvreseq 4772	. . . . . . . . . . . . . . . . . . . . 21 |- (((B Fn On /\ F Fn On) /\ x C_ On) -> ((B |` x) = (F |` x) <-> A.y e. x (B` y) = (F` y)))
15	13, 14	mpanl2 771	. . . . . . . . . . . . . . . . . . . 20 |- ((B Fn On /\ x C_ On) -> ((B |` x) = (F |` x) <-> A.y e. x (B` y) = (F` y)))
16		fveq2 4681	. . . . . . . . . . . . . . . . . . . 20 |- ((B |` x) = (F |` x) -> (G` (B |` x)) = (G` (F |` x)))
17	15, 16	syl6bir 232	. . . . . . . . . . . . . . . . . . 19 |- ((B Fn On /\ x C_ On) -> (A.y e. x (B` y) = (F` y) -> (G` (B |` x)) = (G` (F |` x))))
18		onss 3869	. . . . . . . . . . . . . . . . . . 19 |- (x e. On -> x C_ On)
19	17, 18	sylan2 500	. . . . . . . . . . . . . . . . . 18 |- ((B Fn On /\ x e. On) -> (A.y e. x (B` y) = (F` y) -> (G` (B |` x)) = (G` (F |` x))))
20	19	ancoms 484	. . . . . . . . . . . . . . . . 17 |- ((x e. On /\ B Fn On) -> (A.y e. x (B` y) = (F` y) -> (G` (B |` x)) = (G` (F |` x))))
21	20	imp 377	. . . . . . . . . . . . . . . 16 |- (((x e. On /\ B Fn On) /\ A.y e. x (B` y) = (F` y)) -> (G` (B |` x)) = (G` (F |` x)))
22	21	adantr 425	. . . . . . . . . . . . . . 15 |- ((((x e. On /\ B Fn On) /\ A.y e. x (B` y) = (F` y)) /\ ((x e. On -> (B` x) = (G` (B |` x))) /\ x e. On)) -> (G` (B |` x)) = (G` (F |` x)))
23	11, 12	tfr2 5133	. . . . . . . . . . . . . . . . . . . 20 |- (x e. On -> (F` x) = (G` (F |` x)))
24	23	jctr 315	. . . . . . . . . . . . . . . . . . 19 |- ((x e. On -> (B` x) = (G` (B |` x))) -> ((x e. On -> (B` x) = (G` (B |` x))) /\ (x e. On -> (F` x) = (G` (F |` x)))))
25		jcab 659	. . . . . . . . . . . . . . . . . . 19 |- ((x e. On -> ((B` x) = (G` (B |` x)) /\ (F` x) = (G` (F |` x)))) <-> ((x e. On -> (B` x) = (G` (B |` x))) /\ (x e. On -> (F` x) = (G` (F |` x)))))
26	24, 25	sylibr 217	. . . . . . . . . . . . . . . . . 18 |- ((x e. On -> (B` x) = (G` (B |` x))) -> (x e. On -> ((B` x) = (G` (B |` x)) /\ (F` x) = (G` (F |` x)))))
27		eqeq12 1896	. . . . . . . . . . . . . . . . . 18 |- (((B` x) = (G` (B |` x)) /\ (F` x) = (G` (F |` x))) -> ((B` x) = (F` x) <-> (G` (B |` x)) = (G` (F |` x))))
28	26, 27	syl6 25	. . . . . . . . . . . . . . . . 17 |- ((x e. On -> (B` x) = (G` (B |` x))) -> (x e. On -> ((B` x) = (F` x) <-> (G` (B |` x)) = (G` (F |` x)))))
29	28	imp 377	. . . . . . . . . . . . . . . 16 |- (((x e. On -> (B` x) = (G` (B |` x))) /\ x e. On) -> ((B` x) = (F` x) <-> (G` (B |` x)) = (G` (F |` x))))
30	29	adantl 424	. . . . . . . . . . . . . . 15 |- ((((x e. On /\ B Fn On) /\ A.y e. x (B` y) = (F` y)) /\ ((x e. On -> (B` x) = (G` (B |` x))) /\ x e. On)) -> ((B` x) = (F` x) <-> (G` (B |` x)) = (G` (F |` x))))
31	22, 30	mpbird 213	. . . . . . . . . . . . . 14 |- ((((x e. On /\ B Fn On) /\ A.y e. x (B` y) = (F` y)) /\ ((x e. On -> (B` x) = (G` (B |` x))) /\ x e. On)) -> (B` x) = (F` x))
32	31	exp43 415	. . . . . . . . . . . . 13 |- ((x e. On /\ B Fn On) -> (A.y e. x (B` y) = (F` y) -> ((x e. On -> (B` x) = (G` (B |` x))) -> (x e. On -> (B` x) = (F` x)))))
33	32	com4t 44	. . . . . . . . . . . 12 |- ((x e. On -> (B` x) = (G` (B |` x))) -> (x e. On -> ((x e. On /\ B Fn On) -> (A.y e. x (B` y) = (F` y) -> (B` x) = (F` x)))))
34	33	exp4a 409	. . . . . . . . . . 11 |- ((x e. On -> (B` x) = (G` (B |` x))) -> (x e. On -> (x e. On -> (B Fn On -> (A.y e. x (B` y) = (F` y) -> (B` x) = (F` x))))))
35	34	pm2.43d 79	. . . . . . . . . 10 |- ((x e. On -> (B` x) = (G` (B |` x))) -> (x e. On -> (B Fn On -> (A.y e. x (B` y) = (F` y) -> (B` x) = (F` x)))))
36	10, 35	syl 12	. . . . . . . . 9 |- (A.x e. On (B` x) = (G` (B |` x)) -> (x e. On -> (B Fn On -> (A.y e. x (B` y) = (F` y) -> (B` x) = (F` x)))))
37	36	com3l 38	. . . . . . . 8 |- (x e. On -> (B Fn On -> (A.x e. On (B` x) = (G` (B |` x)) -> (A.y e. x (B` y) = (F` y) -> (B` x) = (F` x)))))
38	37	imp3a 388	. . . . . . 7 |- (x e. On -> ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (A.y e. x (B` y) = (F` y) -> (B` x) = (F` x))))
39	38	a2d 16	. . . . . 6 |- (x e. On -> (((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> A.y e. x (B` y) = (F` y)) -> ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (B` x) = (F` x))))
40		r19.21v 2178	. . . . . 6 |- (A.y e. x ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (B` y) = (F` y)) <-> ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> A.y e. x (B` y) = (F` y)))
41	39, 40	syl5ib 223	. . . . 5 |- (x e. On -> (A.y e. x ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (B` y) = (F` y)) -> ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (B` x) = (F` x))))
42	5, 9, 41	tfis2f 3939	. . . 4 |- (x e. On -> ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (B` x) = (F` x)))
43	42	com12 14	. . 3 |- ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> (x e. On -> (B` x) = (F` x)))
44	3, 43	r19.21ai 2174	. 2 |- ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> A.x e. On (B` x) = (F` x))
45		eqid 1884	. . 3 |- On = On
46		eqfnfv 4766	. . . . 5 |- ((B Fn On /\ F Fn On) -> (B = F <-> (On = On /\ A.x e. On (B` x) = (F` x))))
47	13, 46	mpan2 760	. . . 4 |- (B Fn On -> (B = F <-> (On = On /\ A.x e. On (B` x) = (F` x))))
48	47	biimpar 461	. . 3 |- ((B Fn On /\ (On = On /\ A.x e. On (B` x) = (F` x))) -> B = F)
49	45, 48	mpanr1 774	. 2 |- ((B Fn On /\ A.x e. On (B` x) = (F` x)) -> B = F)
50	44, 49	syldan 516	1 |- ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> B = F)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106   C_ wss 2593  U.cuni 3177  Oncon0 3657   |` cres 3988   Fn wfn 3993  ` cfv 3998

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-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-suc 3663  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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