Theorem seqomlem4 8091

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 7604	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2	1	fvresd 6692	. . . . . 6 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
3		frsuc 8074	. . . . . . . 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 6692	. . . . . . . . 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 6673	. . . . . . . . 9 ⊢ (𝑄‘suc 𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝐴)
7	4, 6	syl6eqr 2876	. . . . . . . 8 ⊢ (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
8		fvres 6691	. . . . . . . . . 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 6673	. . . . . . . . . 10 ⊢ (𝑄‘𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝐴)
10	8, 9	syl6eqr 2876	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴) = (𝑄‘𝐴))
11	10	fveq2d 6676	. . . . . . . 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 2866	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝐴)))
13	5	seqomlem1 8088	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑄‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
14	13	fveq2d 6676	. . . . . . 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 7161	. . . . . . . 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 6685	. . . . . . . . . 10 ⊢ (2nd ‘(𝑄‘𝐴)) ∈ V
17		suceq 6258	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐴 → suc 𝑖 = suc 𝐴)
18		oveq1 7165	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐴 → (𝑖𝐹𝑣) = (𝐴𝐹𝑣))
19	17, 18	opeq12d 4813	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐴 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹𝑣)⟩)
20		oveq2 7166	. . . . . . . . . . . 12 ⊢ (𝑣 = (2nd ‘(𝑄‘𝐴)) → (𝐴𝐹𝑣) = (𝐴𝐹(2nd ‘(𝑄‘𝐴))))
21	20	opeq2d 4812	. . . . . . . . . . 11 ⊢ (𝑣 = (2nd ‘(𝑄‘𝐴)) → ⟨suc 𝐴, (𝐴𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩)
22		eqid 2823	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
23		opex 5358	. . . . . . . . . . 11 ⊢ ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩ ∈ V
24	19, 21, 22, 23	ovmpo 7312	. . . . . . . . . 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 689	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩)
26		fvres 6691	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = (𝑄‘𝐴))
27	26, 13	eqtrd 2858	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
28		frfnom 8072	. . . . . . . . . . . . . . . . . 18 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
29	5	reseq1i 5851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
30	29	fneq1i 6452	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
31	28, 30	mpbir 233	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 ↾ ω) Fn ω
32		fnfvelrn 6850	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄 ↾ ω) Fn ω ∧ 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
33	31, 32	mpan 688	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
34	27, 33	eqeltrrd 2916	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
35		df-ima 5570	. . . . . . . . . . . . . . 15 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω)
36	34, 35	eleqtrrdi 2926	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩ ∈ (𝑄 “ ω))
37		df-br 5069	. . . . . . . . . . . . . 14 ⊢ (𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴)) ↔ ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩ ∈ (𝑄 “ ω))
38	36, 37	sylibr 236	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → 𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴)))
39	5	seqomlem2 8089	. . . . . . . . . . . . . 14 ⊢ (𝑄 “ ω) Fn ω
40		fnbrfvb 6720	. . . . . . . . . . . . . 14 ⊢ (((𝑄 “ ω) Fn ω ∧ 𝐴 ∈ ω) → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄‘𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴))))
41	39, 40	mpan 688	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄‘𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴))))
42	38, 41	mpbird 259	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → ((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄‘𝐴)))
43	42	eqcomd 2829	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (2nd ‘(𝑄‘𝐴)) = ((𝑄 “ ω)‘𝐴))
44	43	oveq2d 7174	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴𝐹(2nd ‘(𝑄‘𝐴))) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
45	44	opeq2d 4812	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩ = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
46	25, 45	eqtrd 2858	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝐴))) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
47	15, 46	syl5eqr 2872	. . . . . . 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 2862	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
49	2, 48	eqtrd 2858	. . . . 5 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
50		fnfvelrn 6850	. . . . . 6 ⊢ (((𝑄 ↾ ω) Fn ω ∧ suc 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
51	31, 1, 50	sylancr 589	. . . . 5 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
52	49, 51	eqeltrrd 2916	. . . 4 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
53	52, 35	eleqtrrdi 2926	. . 3 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
54		df-br 5069	. . 3 ⊢ (suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
55	53, 54	sylibr 236	. 2 ⊢ (𝐴 ∈ ω → suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)))
56		fnbrfvb 6720	. . 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 589	. 2 ⊢ (𝐴 ∈ ω → (((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴))))
58	55, 57	mpbird 259	1 ⊢ (𝐴 ∈ ω → ((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ∅c0 4293  ⟨cop 4575   class class class wbr 5068   I cid 5461  ran crn 5558   ↾ cres 5559   “ cima 5560  suc csuc 6195   Fn wfn 6352  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  ωcom 7582  2^nd c2nd 7690  reccrdg 8047

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048

This theorem is referenced by:  seqomsuc  8095