Theorem tfrlem1 5119

Description: A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47.

Assertion

Ref	Expression
tfrlem1	|- (A e. On -> ((F Fn A /\ G Fn A) -> (A.x e. A ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. A (F` x) = (G` x))))

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

Proof of Theorem tfrlem1

Step	Hyp	Ref	Expression
1		ssid 2634	. 2 |- A C_ A
2		sseq1 2637	. . . . 5 |- (y = A -> (y C_ A <-> A C_ A))
3		raleq 2266	. . . . . 6 |- (y = A -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) <-> A.x e. A ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))))
4		raleq 2266	. . . . . 6 |- (y = A -> (A.x e. y (F` x) = (G` x) <-> A.x e. A (F` x) = (G` x)))
5	3, 4	imbi12d 688	. . . . 5 |- (y = A -> ((A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. y (F` x) = (G` x)) <-> (A.x e. A ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. A (F` x) = (G` x))))
6	2, 5	imbi12d 688	. . . 4 |- (y = A -> ((y C_ A -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. y (F` x) = (G` x))) <-> (A C_ A -> (A.x e. A ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. A (F` x) = (G` x)))))
7	6	imbi2d 674	. . 3 |- (y = A -> (((F Fn A /\ G Fn A) -> (y C_ A -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. y (F` x) = (G` x)))) <-> ((F Fn A /\ G Fn A) -> (A C_ A -> (A.x e. A ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. A (F` x) = (G` x))))))
8		sseq1 2637	. . . . . . . 8 |- (y = z -> (y C_ A <-> z C_ A))
9	8	anbi2d 678	. . . . . . 7 |- (y = z -> (((F Fn A /\ G Fn A) /\ y C_ A) <-> ((F Fn A /\ G Fn A) /\ z C_ A)))
10		raleq 2266	. . . . . . 7 |- (y = z -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) <-> A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))))
11	9, 10	anbi12d 690	. . . . . 6 |- (y = z -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) <-> (((F Fn A /\ G Fn A) /\ z C_ A) /\ A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))))))
12		raleq 2266	. . . . . 6 |- (y = z -> (A.x e. y (F` x) = (G` x) <-> A.x e. z (F` x) = (G` x)))
13	11, 12	imbi12d 688	. . . . 5 |- (y = z -> (((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.x e. y (F` x) = (G` x)) <-> ((((F Fn A /\ G Fn A) /\ z C_ A) /\ A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.x e. z (F` x) = (G` x))))
14		onelss 3705	. . . . . . . . . . 11 |- (y e. On -> (z e. y -> z C_ y))
15		sstr2 2623	. . . . . . . . . . . . 13 |- (z C_ y -> (y C_ A -> z C_ A))
16	15	anim2d 620	. . . . . . . . . . . 12 |- (z C_ y -> (((F Fn A /\ G Fn A) /\ y C_ A) -> ((F Fn A /\ G Fn A) /\ z C_ A)))
17		ax-17 1317	. . . . . . . . . . . . 13 |- (z C_ y -> A.x z C_ y)
18		hbra1 2147	. . . . . . . . . . . . 13 |- (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.xA.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))))
19		ssel 2615	. . . . . . . . . . . . . 14 |- (z C_ y -> (x e. z -> x e. y))
20		ra4 2155	. . . . . . . . . . . . . 14 |- (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> (x e. y -> ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))))
21	19, 20	syl9 71	. . . . . . . . . . . . 13 |- (z C_ y -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> (x e. z -> ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))))))
22	17, 18, 21	r19.21ad 2180	. . . . . . . . . . . 12 |- (z C_ y -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))))
23	16, 22	anim12d 617	. . . . . . . . . . 11 |- (z C_ y -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> (((F Fn A /\ G Fn A) /\ z C_ A) /\ A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))))))
24	14, 23	syl6 25	. . . . . . . . . 10 |- (y e. On -> (z e. y -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> (((F Fn A /\ G Fn A) /\ z C_ A) /\ A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))))))
25	24	com23 36	. . . . . . . . 9 |- (y e. On -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> (z e. y -> (((F Fn A /\ G Fn A) /\ z C_ A) /\ A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))))))
26	25	r19.21adv 2181	. . . . . . . 8 |- (y e. On -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.z e. y (((F Fn A /\ G Fn A) /\ z C_ A) /\ A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))))))
27		onelss 3705	. . . . . . . . . . . . . . . . . 18 |- (y e. On -> (x e. y -> x C_ y))
28		sstr2 2623	. . . . . . . . . . . . . . . . . . 19 |- (x C_ y -> (y C_ A -> x C_ A))
29		eqeq12 1896	. . . . . . . . . . . . . . . . . . . . . 22 |- (((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> ((F` x) = (G` x) <-> (B` (F |` x)) = (B` (G |` x))))
30		fvreseq 4772	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((F Fn A /\ G Fn A) /\ x C_ A) -> ((F |` x) = (G |` x) <-> A.w e. x (F` w) = (G` w)))
31	30	biimpar 461	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((F Fn A /\ G Fn A) /\ x C_ A) /\ A.w e. x (F` w) = (G` w)) -> (F |` x) = (G |` x))
32	31	fveq2d 4685	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((F Fn A /\ G Fn A) /\ x C_ A) /\ A.w e. x (F` w) = (G` w)) -> (B` (F |` x)) = (B` (G |` x)))
33	29, 32	syl5bir 227	. . . . . . . . . . . . . . . . . . . . 21 |- (((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> ((((F Fn A /\ G Fn A) /\ x C_ A) /\ A.w e. x (F` w) = (G` w)) -> (F` x) = (G` x)))
34	33	exp4c 411	. . . . . . . . . . . . . . . . . . . 20 |- (((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> ((F Fn A /\ G Fn A) -> (x C_ A -> (A.w e. x (F` w) = (G` w) -> (F` x) = (G` x)))))
35	34	com4l 43	. . . . . . . . . . . . . . . . . . 19 |- ((F Fn A /\ G Fn A) -> (x C_ A -> (A.w e. x (F` w) = (G` w) -> (((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> (F` x) = (G` x)))))
36	28, 35	syl9 71	. . . . . . . . . . . . . . . . . 18 |- (x C_ y -> ((F Fn A /\ G Fn A) -> (y C_ A -> (A.w e. x (F` w) = (G` w) -> (((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> (F` x) = (G` x))))))
37	27, 36	syl6 25	. . . . . . . . . . . . . . . . 17 |- (y e. On -> (x e. y -> ((F Fn A /\ G Fn A) -> (y C_ A -> (A.w e. x (F` w) = (G` w) -> (((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> (F` x) = (G` x)))))))
38	37	imp4a 391	. . . . . . . . . . . . . . . 16 |- (y e. On -> (x e. y -> (((F Fn A /\ G Fn A) /\ y C_ A) -> (A.w e. x (F` w) = (G` w) -> (((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> (F` x) = (G` x))))))
39	38	com23 36	. . . . . . . . . . . . . . 15 |- (y e. On -> (((F Fn A /\ G Fn A) /\ y C_ A) -> (x e. y -> (A.w e. x (F` w) = (G` w) -> (((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> (F` x) = (G` x))))))
40	39	imp31 389	. . . . . . . . . . . . . 14 |- (((y e. On /\ ((F Fn A /\ G Fn A) /\ y C_ A)) /\ x e. y) -> (A.w e. x (F` w) = (G` w) -> (((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> (F` x) = (G` x))))
41	40	ralimdvaa 2171	. . . . . . . . . . . . 13 |- ((y e. On /\ ((F Fn A /\ G Fn A) /\ y C_ A)) -> (A.x e. y A.w e. x (F` w) = (G` w) -> A.x e. y (((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> (F` x) = (G` x))))
42		ralim 2164	. . . . . . . . . . . . 13 |- (A.x e. y (((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> (F` x) = (G` x)) -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. y (F` x) = (G` x)))
43	41, 42	syl6 25	. . . . . . . . . . . 12 |- ((y e. On /\ ((F Fn A /\ G Fn A) /\ y C_ A)) -> (A.x e. y A.w e. x (F` w) = (G` w) -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. y (F` x) = (G` x))))
44		hbra1 2147	. . . . . . . . . . . . 13 |- (A.x e. z (F` x) = (G` x) -> A.xA.x e. z (F` x) = (G` x))
45		ax-17 1317	. . . . . . . . . . . . 13 |- (A.w e. x (F` w) = (G` w) -> A.zA.w e. x (F` w) = (G` w))
46		raleq 2266	. . . . . . . . . . . . . 14 |- (z = x -> (A.w e. z (F` w) = (G` w) <-> A.w e. x (F` w) = (G` w)))
47		fveq2 4681	. . . . . . . . . . . . . . . 16 |- (x = w -> (F` x) = (F` w))
48		fveq2 4681	. . . . . . . . . . . . . . . 16 |- (x = w -> (G` x) = (G` w))
49	47, 48	eqeq12d 1899	. . . . . . . . . . . . . . 15 |- (x = w -> ((F` x) = (G` x) <-> (F` w) = (G` w)))
50	49	cbvralv 2280	. . . . . . . . . . . . . 14 |- (A.x e. z (F` x) = (G` x) <-> A.w e. z (F` w) = (G` w))
51	46, 50	syl5bb 591	. . . . . . . . . . . . 13 |- (z = x -> (A.x e. z (F` x) = (G` x) <-> A.w e. x (F` w) = (G` w)))
52	44, 45, 51	cbvral 2278	. . . . . . . . . . . 12 |- (A.z e. y A.x e. z (F` x) = (G` x) <-> A.x e. y A.w e. x (F` w) = (G` w))
53	43, 52	syl5ib 223	. . . . . . . . . . 11 |- ((y e. On /\ ((F Fn A /\ G Fn A) /\ y C_ A)) -> (A.z e. y A.x e. z (F` x) = (G` x) -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. y (F` x) = (G` x))))
54	53	ex 402	. . . . . . . . . 10 |- (y e. On -> (((F Fn A /\ G Fn A) /\ y C_ A) -> (A.z e. y A.x e. z (F` x) = (G` x) -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. y (F` x) = (G` x)))))
55	54	com23 36	. . . . . . . . 9 |- (y e. On -> (A.z e. y A.x e. z (F` x) = (G` x) -> (((F Fn A /\ G Fn A) /\ y C_ A) -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. y (F` x) = (G` x)))))
56	55	imp4a 391	. . . . . . . 8 |- (y e. On -> (A.z e. y A.x e. z (F` x) = (G` x) -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.x e. y (F` x) = (G` x))))
57	26, 56	imim12d 69	. . . . . . 7 |- (y e. On -> ((A.z e. y (((F Fn A /\ G Fn A) /\ z C_ A) /\ A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.z e. y A.x e. z (F` x) = (G` x)) -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.x e. y (F` x) = (G` x)))))
58		pm2.43 77	. . . . . . 7 |- (((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.x e. y (F` x) = (G` x))) -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.x e. y (F` x) = (G` x)))
59	57, 58	syl6 25	. . . . . 6 |- (y e. On -> ((A.z e. y (((F Fn A /\ G Fn A) /\ z C_ A) /\ A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.z e. y A.x e. z (F` x) = (G` x)) -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.x e. y (F` x) = (G` x))))
60		ralim 2164	. . . . . 6 |- (A.z e. y ((((F Fn A /\ G Fn A) /\ z C_ A) /\ A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.x e. z (F` x) = (G` x)) -> (A.z e. y (((F Fn A /\ G Fn A) /\ z C_ A) /\ A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.z e. y A.x e. z (F` x) = (G` x)))
61	59, 60	syl5 20	. . . . 5 |- (y e. On -> (A.z e. y ((((F Fn A /\ G Fn A) /\ z C_ A) /\ A.x e. z ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.x e. z (F` x) = (G` x)) -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.x e. y (F` x) = (G` x))))
62	13, 61	tfis2 3940	. . . 4 |- (y e. On -> ((((F Fn A /\ G Fn A) /\ y C_ A) /\ A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x)))) -> A.x e. y (F` x) = (G` x)))
63	62	exp4c 411	. . 3 |- (y e. On -> ((F Fn A /\ G Fn A) -> (y C_ A -> (A.x e. y ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. y (F` x) = (G` x)))))
64	7, 63	vtoclga 2352	. 2 |- (A e. On -> ((F Fn A /\ G Fn A) -> (A C_ A -> (A.x e. A ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. A (F` x) = (G` x)))))
65	1, 64	mpii 56	1 |- (A e. On -> ((F Fn A /\ G Fn A) -> (A.x e. A ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. A (F` x) = (G` x))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  Oncon0 3657   |` cres 3988   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  tfrlem2 5120

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-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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