Theorem seqomlem4 8438

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 7884	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2	1	fvresd 6901	. . . . . 6 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
3		frsuc 8422	. . . . . . . 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 6901	. . . . . . . . 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 6882	. . . . . . . . 9 ⊢ (𝑄‘suc 𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝐴)
7	4, 6	eqtr4di 2815	. . . . . . . 8 ⊢ (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
8		fvres 6900	. . . . . . . . . 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 6882	. . . . . . . . . 10 ⊢ (𝑄‘𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝐴)
10	8, 9	eqtr4di 2815	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴) = (𝑄‘𝐴))
11	10	fveq2d 6885	. . . . . . . 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 2805	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄‘𝐴)))
13	5	seqomlem1 8435	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑄‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
14	13	fveq2d 6885	. . . . . . 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 7415	. . . . . . . 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 6894	. . . . . . . . . 10 ⊢ (2nd ‘(𝑄‘𝐴)) ∈ V
17		suceq 6429	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐴 → suc 𝑖 = suc 𝐴)
18		oveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐴 → (𝑖𝐹𝑣) = (𝐴𝐹𝑣))
19	17, 18	opeq12d 4845	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐴 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹𝑣)⟩)
20		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑣 = (2nd ‘(𝑄‘𝐴)) → (𝐴𝐹𝑣) = (𝐴𝐹(2nd ‘(𝑄‘𝐴))))
21	20	opeq2d 4844	. . . . . . . . . . 11 ⊢ (𝑣 = (2nd ‘(𝑄‘𝐴)) → ⟨suc 𝐴, (𝐴𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩)
22		eqid 2762	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
23		opex 5444	. . . . . . . . . . 11 ⊢ ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩ ∈ V
24	19, 21, 22, 23	ovmpo 7572	. . . . . . . . . 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 703	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩)
26		fvres 6900	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = (𝑄‘𝐴))
27	26, 13	eqtrd 2797	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
28		frfnom 8420	. . . . . . . . . . . . . . . . . 18 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
29	5	reseq1i 5973	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
30	29	fneq1i 6632	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
31	28, 30	mpbir 234	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 ↾ ω) Fn ω
32		fnfvelrn 7075	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄 ↾ ω) Fn ω ∧ 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
33	31, 32	mpan 702	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
34	27, 33	eqeltrrd 2863	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
35		df-ima 5673	. . . . . . . . . . . . . . 15 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω)
36	34, 35	eleqtrrdi 2873	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩ ∈ (𝑄 “ ω))
37		df-br 5109	. . . . . . . . . . . . . 14 ⊢ (𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴)) ↔ ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩ ∈ (𝑄 “ ω))
38	36, 37	sylibr 237	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → 𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴)))
39	5	seqomlem2 8436	. . . . . . . . . . . . . 14 ⊢ (𝑄 “ ω) Fn ω
40		fnbrfvb 6931	. . . . . . . . . . . . . 14 ⊢ (((𝑄 “ ω) Fn ω ∧ 𝐴 ∈ ω) → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄‘𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴))))
41	39, 40	mpan 702	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄‘𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄‘𝐴))))
42	38, 41	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → ((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄‘𝐴)))
43	42	eqcomd 2768	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (2nd ‘(𝑄‘𝐴)) = ((𝑄 “ ω)‘𝐴))
44	43	oveq2d 7428	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴𝐹(2nd ‘(𝑄‘𝐴))) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
45	44	opeq2d 4844	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄‘𝐴)))⟩ = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
46	25, 45	eqtrd 2797	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄‘𝐴))) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
47	15, 46	eqtr3id 2811	. . . . . . 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 2801	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
49	2, 48	eqtrd 2797	. . . . 5 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
50		fnfvelrn 7075	. . . . . 6 ⊢ (((𝑄 ↾ ω) Fn ω ∧ suc 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
51	31, 1, 50	sylancr 598	. . . . 5 ⊢ (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
52	49, 51	eqeltrrd 2863	. . . 4 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
53	52, 35	eleqtrrdi 2873	. . 3 ⊢ (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
54		df-br 5109	. . 3 ⊢ (suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
55	53, 54	sylibr 237	. 2 ⊢ (𝐴 ∈ ω → suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)))
56		fnbrfvb 6931	. . 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 598	. 2 ⊢ (𝐴 ∈ ω → (((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴))))
58	55, 57	mpbird 260	1 ⊢ (𝐴 ∈ ω → ((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  ⟨cop 4594   class class class wbr 5108   I cid 5554  ran crn 5661   ↾ cres 5662   “ cima 5663  suc csuc 6362   Fn wfn 6531  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414  ωcom 7860  2^nd c2nd 7983  reccrdg 8394

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395

This theorem is used by:  seqomsuc  8442