Theorem seqomlem4 8384

Description: Lemma for seq_ω. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)

Hypothesis

Ref	Expression
seqomlem.a	⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)

Assertion

Ref	Expression
seqomlem4	⊢ (𝐴 ∈ ω → ((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))

Distinct variable groups:   𝑄,𝑖,𝑣   𝐴,𝑖,𝑣   𝑖,𝐹,𝑣

Allowed substitution hints:   𝐼(𝑣,𝑖)

Proof of Theorem seqomlem4

Step	Hyp	Ref	Expression
1		peano2 7832	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2	1	fvresd 6854	. . . . . 6 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
3		frsuc 8368	. . . . . . . 8 ⊢ (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝐴) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴)))
4	1	fvresd 6854	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝐴))
5		seqomlem.a	. . . . . . . . . 10 ⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)
6	5	fveq1i 6835	. . . . . . . . 9 ⊢ (𝑄‘suc 𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝐴)
7	4, 6	eqtr4di 2789	. . . . . . . 8 ⊢ (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
8		fvres 6853	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝐴))
9	5	fveq1i 6835	. . . . . . . . . 10 ⊢ (𝑄‘𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝐴)
10	8, 9	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴) = (𝑄‘𝐴))
11	10	fveq2d 6838	. . . . . . . 8 ⊢ (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴)) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝐴)))
12	3, 7, 11	3eqtr3d 2779	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝐴)))
13	5	seqomlem1 8381	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑄‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
14	13	fveq2d 6838	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝐴)) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩))
15		df-ov 7361	. . . . . . . 8 ⊢ (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝐴))) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
16		fvex 6847	. . . . . . . . . 10 ⊢ (2nd ‘(𝑄‘𝐴)) ∈ V
17		suceq 6385	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐴 → suc 𝑖 = suc 𝐴)
18		oveq1 7365	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐴 → (𝑖𝐹𝑣) = (𝐴𝐹𝑣))
19	17, 18	opeq12d 4837	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐴 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹𝑣)⟩)
20		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝑣 = (2nd ‘(𝑄‘𝐴)) → (𝐴𝐹𝑣) = (𝐴𝐹(2nd ‘(𝑄‘𝐴))))
21	20	opeq2d 4836	. . . . . . . . . . 11 ⊢ (𝑣 = (2nd ‘(𝑄‘𝐴)) → ⟨suc 𝐴, (𝐴𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩)
22		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
23		opex 5412	. . . . . . . . . . 11 ⊢ ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩ ∈ V
24	19, 21, 22, 23	ovmpo 7518	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ (2nd ‘(𝑄‘𝐴)) ∈ V) → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩)
25	16, 24	mpan2 691	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩)
26		fvres 6853	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = (𝑄‘𝐴))
27	26, 13	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
28		frfnom 8366	. . . . . . . . . . . . . . . . . 18 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
29	5	reseq1i 5934	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
30	29	fneq1i 6589	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
31	28, 30	mpbir 231	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 ↾ ω) Fn ω
32		fnfvelrn 7025	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄 ↾ ω) Fn ω ∧ 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
33	31, 32	mpan 690	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
34	27, 33	eqeltrrd 2837	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
35		df-ima 5637	. . . . . . . . . . . . . . 15 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω)
36	34, 35	eleqtrrdi 2847	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩ ∈ (𝑄 “ ω))
37		df-br 5099	. . . . . . . . . . . . . 14 ⊢ (𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴)) ↔ ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩ ∈ (𝑄 “ ω))
38	36, 37	sylibr 234	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → 𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴)))
39	5	seqomlem2 8382	. . . . . . . . . . . . . 14 ⊢ (𝑄 “ ω) Fn ω
40		fnbrfvb 6884	. . . . . . . . . . . . . 14 ⊢ (((𝑄 “ ω) Fn ω ∧ 𝐴 ∈ ω) → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄‘𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴))))
41	39, 40	mpan 690	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄‘𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴))))
42	38, 41	mpbird 257	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → ((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄‘𝐴)))
43	42	eqcomd 2742	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (2nd ‘(𝑄‘𝐴)) = ((𝑄 “ ω)‘𝐴))
44	43	oveq2d 7374	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴𝐹(2nd ‘(𝑄‘𝐴))) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
45	44	opeq2d 4836	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩ = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
46	25, 45	eqtrd 2771	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝐴))) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
47	15, 46	eqtr3id 2785	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
48	12, 14, 47	3eqtrd 2775	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
49	2, 48	eqtrd 2771	. . . . 5 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
50		fnfvelrn 7025	. . . . . 6 ⊢ (((𝑄 ↾ ω) Fn ω ∧ suc 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
51	31, 1, 50	sylancr 587	. . . . 5 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
52	49, 51	eqeltrrd 2837	. . . 4 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
53	52, 35	eleqtrrdi 2847	. . 3 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
54		df-br 5099	. . 3 ⊢ (suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
55	53, 54	sylibr 234	. 2 ⊢ (𝐴 ∈ ω → suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)))
56		fnbrfvb 6884	. . 3 ⊢ (((𝑄 “ ω) Fn ω ∧ suc 𝐴 ∈ ω) → (((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴))))
57	39, 1, 56	sylancr 587	. 2 ⊢ (𝐴 ∈ ω → (((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴))))
58	55, 57	mpbird 257	1 ⊢ (𝐴 ∈ ω → ((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  ⟨cop 4586   class class class wbr 5098   I cid 5518  ran crn 5625   ↾ cres 5626   “ cima 5627  suc csuc 6319   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  ωcom 7808  2^nd c2nd 7932  reccrdg 8340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341

This theorem is referenced by:  seqomsuc  8388