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Theorem seqomlem4 8455
Description: Lemma for seqω. (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
StepHypRef Expression
1 peano2 7883 . . . . . . 7 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
21fvresd 6910 . . . . . 6 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
3 frsuc 8439 . . . . . . . 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)))
41fvresd 6910 . . . . . . . . 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)⟩)
65fveq1i 6891 . . . . . . . . 9 (𝑄‘suc 𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝐴)
74, 6eqtr4di 2788 . . . . . . . 8 (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
8 fvres 6909 . . . . . . . . . 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))
95fveq1i 6891 . . . . . . . . . 10 (𝑄𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝐴)
108, 9eqtr4di 2788 . . . . . . . . 9 (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴) = (𝑄𝐴))
1110fveq2d 6894 . . . . . . . 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)⟩)‘(𝑄𝐴)))
123, 7, 113eqtr3d 2778 . . . . . . 7 (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝐴)))
135seqomlem1 8452 . . . . . . . 8 (𝐴 ∈ ω → (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
1413fveq2d 6894 . . . . . . 7 (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝐴)) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩))
15 df-ov 7414 . . . . . . . 8 (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
16 fvex 6903 . . . . . . . . . 10 (2nd ‘(𝑄𝐴)) ∈ V
17 suceq 6429 . . . . . . . . . . . 12 (𝑖 = 𝐴 → suc 𝑖 = suc 𝐴)
18 oveq1 7418 . . . . . . . . . . . 12 (𝑖 = 𝐴 → (𝑖𝐹𝑣) = (𝐴𝐹𝑣))
1917, 18opeq12d 4880 . . . . . . . . . . 11 (𝑖 = 𝐴 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹𝑣)⟩)
20 oveq2 7419 . . . . . . . . . . . 12 (𝑣 = (2nd ‘(𝑄𝐴)) → (𝐴𝐹𝑣) = (𝐴𝐹(2nd ‘(𝑄𝐴))))
2120opeq2d 4879 . . . . . . . . . . 11 (𝑣 = (2nd ‘(𝑄𝐴)) → ⟨suc 𝐴, (𝐴𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
22 eqid 2730 . . . . . . . . . . 11 (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
23 opex 5463 . . . . . . . . . . 11 ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩ ∈ V
2419, 21, 22, 23ovmpo 7570 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ (2nd ‘(𝑄𝐴)) ∈ V) → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
2516, 24mpan2 687 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
26 fvres 6909 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = (𝑄𝐴))
2726, 13eqtrd 2770 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
28 frfnom 8437 . . . . . . . . . . . . . . . . . 18 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
295reseq1i 5976 . . . . . . . . . . . . . . . . . . 19 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
3029fneq1i 6645 . . . . . . . . . . . . . . . . . 18 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
3128, 30mpbir 230 . . . . . . . . . . . . . . . . 17 (𝑄 ↾ ω) Fn ω
32 fnfvelrn 7081 . . . . . . . . . . . . . . . . 17 (((𝑄 ↾ ω) Fn ω ∧ 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
3331, 32mpan 686 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
3427, 33eqeltrrd 2832 . . . . . . . . . . . . . . 15 (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ ran (𝑄 ↾ ω))
35 df-ima 5688 . . . . . . . . . . . . . . 15 (𝑄 “ ω) = ran (𝑄 ↾ ω)
3634, 35eleqtrrdi 2842 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ (𝑄 “ ω))
37 df-br 5148 . . . . . . . . . . . . . 14 (𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴)) ↔ ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ (𝑄 “ ω))
3836, 37sylibr 233 . . . . . . . . . . . . 13 (𝐴 ∈ ω → 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴)))
395seqomlem2 8453 . . . . . . . . . . . . . 14 (𝑄 “ ω) Fn ω
40 fnbrfvb 6943 . . . . . . . . . . . . . 14 (((𝑄 “ ω) Fn ω ∧ 𝐴 ∈ ω) → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴))))
4139, 40mpan 686 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴))))
4238, 41mpbird 256 . . . . . . . . . . . 12 (𝐴 ∈ ω → ((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)))
4342eqcomd 2736 . . . . . . . . . . 11 (𝐴 ∈ ω → (2nd ‘(𝑄𝐴)) = ((𝑄 “ ω)‘𝐴))
4443oveq2d 7427 . . . . . . . . . 10 (𝐴 ∈ ω → (𝐴𝐹(2nd ‘(𝑄𝐴))) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
4544opeq2d 4879 . . . . . . . . 9 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩ = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4625, 45eqtrd 2770 . . . . . . . 8 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4715, 46eqtr3id 2784 . . . . . . 7 (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4812, 14, 473eqtrd 2774 . . . . . 6 (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
492, 48eqtrd 2770 . . . . 5 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
50 fnfvelrn 7081 . . . . . 6 (((𝑄 ↾ ω) Fn ω ∧ suc 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
5131, 1, 50sylancr 585 . . . . 5 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
5249, 51eqeltrrd 2832 . . . 4 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
5352, 35eleqtrrdi 2842 . . 3 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
54 df-br 5148 . . 3 (suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
5553, 54sylibr 233 . 2 (𝐴 ∈ ω → suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)))
56 fnbrfvb 6943 . . 3 (((𝑄 “ ω) Fn ω ∧ suc 𝐴 ∈ ω) → (((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴))))
5739, 1, 56sylancr 585 . 2 (𝐴 ∈ ω → (((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴))))
5855, 57mpbird 256 1 (𝐴 ∈ ω → ((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2104  Vcvv 3472  c0 4321  cop 4633   class class class wbr 5147   I cid 5572  ran crn 5676  cres 5677  cima 5678  suc csuc 6365   Fn wfn 6537  cfv 6542  (class class class)co 7411  cmpo 7413  ωcom 7857  2nd c2nd 7976  reccrdg 8411
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412
This theorem is referenced by:  seqomsuc  8459
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