Theorem seqomlem4 8430

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 7875	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2	1	fvresd 6885	. . . . . 6 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
3		frsuc 8414	. . . . . . . 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 6885	. . . . . . . . 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 6866	. . . . . . . . 9 ⊢ (𝑄‘suc 𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝐴)
7	4, 6	eqtr4di 2783	. . . . . . . 8 ⊢ (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
8		fvres 6884	. . . . . . . . . 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 6866	. . . . . . . . . 10 ⊢ (𝑄‘𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝐴)
10	8, 9	eqtr4di 2783	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴) = (𝑄‘𝐴))
11	10	fveq2d 6869	. . . . . . . 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 2773	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝐴)))
13	5	seqomlem1 8427	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑄‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
14	13	fveq2d 6869	. . . . . . 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 7397	. . . . . . . 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 6878	. . . . . . . . . 10 ⊢ (2nd ‘(𝑄‘𝐴)) ∈ V
17		suceq 6407	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐴 → suc 𝑖 = suc 𝐴)
18		oveq1 7401	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐴 → (𝑖𝐹𝑣) = (𝐴𝐹𝑣))
19	17, 18	opeq12d 4853	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐴 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹𝑣)⟩)
20		oveq2 7402	. . . . . . . . . . . 12 ⊢ (𝑣 = (2nd ‘(𝑄‘𝐴)) → (𝐴𝐹𝑣) = (𝐴𝐹(2nd ‘(𝑄‘𝐴))))
21	20	opeq2d 4852	. . . . . . . . . . 11 ⊢ (𝑣 = (2nd ‘(𝑄‘𝐴)) → ⟨suc 𝐴, (𝐴𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩)
22		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
23		opex 5432	. . . . . . . . . . 11 ⊢ ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩ ∈ V
24	19, 21, 22, 23	ovmpo 7556	. . . . . . . . . 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 6884	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = (𝑄‘𝐴))
27	26, 13	eqtrd 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
28		frfnom 8412	. . . . . . . . . . . . . . . . . 18 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
29	5	reseq1i 5954	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
30	29	fneq1i 6623	. . . . . . . . . . . . . . . . . 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 7059	. . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
35		df-ima 5659	. . . . . . . . . . . . . . 15 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω)
36	34, 35	eleqtrrdi 2840	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩ ∈ (𝑄 “ ω))
37		df-br 5116	. . . . . . . . . . . . . 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 8428	. . . . . . . . . . . . . 14 ⊢ (𝑄 “ ω) Fn ω
40		fnbrfvb 6918	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (2nd ‘(𝑄‘𝐴)) = ((𝑄 “ ω)‘𝐴))
44	43	oveq2d 7410	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴𝐹(2nd ‘(𝑄‘𝐴))) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
45	44	opeq2d 4852	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩ = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
46	25, 45	eqtrd 2765	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝐴))) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
47	15, 46	eqtr3id 2779	. . . . . . 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 2769	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
49	2, 48	eqtrd 2765	. . . . 5 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
50		fnfvelrn 7059	. . . . . 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 2830	. . . 4 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
53	52, 35	eleqtrrdi 2840	. . 3 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
54		df-br 5116	. . 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 6918	. . 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 1540   ∈ wcel 2109  Vcvv 3455  ∅c0 4304  ⟨cop 4603   class class class wbr 5115   I cid 5540  ran crn 5647   ↾ cres 5648   “ cima 5649  suc csuc 6342   Fn wfn 6514  ‘cfv 6519  (class class class)co 7394   ∈ cmpo 7396  ωcom 7850  2^nd c2nd 7976  reccrdg 8386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5541  df-eprel 5546  df-po 5554  df-so 5555  df-fr 5599  df-we 5601  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7851  df-2nd 7978  df-frecs 8269  df-wrecs 8300  df-recs 8349  df-rdg 8387

This theorem is referenced by:  seqomsuc  8434