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Theorem seqomlem4 7718
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 7252 . . . . . . 7 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2 fvres 6369 . . . . . . 7 (suc 𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
31, 2syl 17 . . . . . 6 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
4 frsuc 7702 . . . . . . . 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)))
5 fvres 6369 . . . . . . . . . 10 (suc 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))
61, 5syl 17 . . . . . . . . 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))
7 seqomlem.a . . . . . . . . . 10 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)
87fveq1i 6354 . . . . . . . . 9 (𝑄‘suc 𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝐴)
96, 8syl6eqr 2812 . . . . . . . 8 (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
10 fvres 6369 . . . . . . . . . 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))
117fveq1i 6354 . . . . . . . . . 10 (𝑄𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝐴)
1210, 11syl6eqr 2812 . . . . . . . . 9 (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴) = (𝑄𝐴))
1312fveq2d 6357 . . . . . . . 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)))
144, 9, 133eqtr3d 2802 . . . . . . 7 (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝐴)))
157seqomlem1 7715 . . . . . . . 8 (𝐴 ∈ ω → (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
1615fveq2d 6357 . . . . . . 7 (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝐴)) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩))
17 df-ov 6817 . . . . . . . 8 (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
18 fvex 6363 . . . . . . . . . 10 (2nd ‘(𝑄𝐴)) ∈ V
19 suceq 5951 . . . . . . . . . . . 12 (𝑖 = 𝐴 → suc 𝑖 = suc 𝐴)
20 oveq1 6821 . . . . . . . . . . . 12 (𝑖 = 𝐴 → (𝑖𝐹𝑣) = (𝐴𝐹𝑣))
2119, 20opeq12d 4561 . . . . . . . . . . 11 (𝑖 = 𝐴 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹𝑣)⟩)
22 oveq2 6822 . . . . . . . . . . . 12 (𝑣 = (2nd ‘(𝑄𝐴)) → (𝐴𝐹𝑣) = (𝐴𝐹(2nd ‘(𝑄𝐴))))
2322opeq2d 4560 . . . . . . . . . . 11 (𝑣 = (2nd ‘(𝑄𝐴)) → ⟨suc 𝐴, (𝐴𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
24 eqid 2760 . . . . . . . . . . 11 (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
25 opex 5081 . . . . . . . . . . 11 ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩ ∈ V
2621, 23, 24, 25ovmpt2 6962 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ (2nd ‘(𝑄𝐴)) ∈ V) → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
2718, 26mpan2 709 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
28 fvres 6369 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = (𝑄𝐴))
2928, 15eqtrd 2794 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
30 frfnom 7700 . . . . . . . . . . . . . . . . . 18 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
317reseq1i 5547 . . . . . . . . . . . . . . . . . . 19 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
3231fneq1i 6146 . . . . . . . . . . . . . . . . . 18 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
3330, 32mpbir 221 . . . . . . . . . . . . . . . . 17 (𝑄 ↾ ω) Fn ω
34 fnfvelrn 6520 . . . . . . . . . . . . . . . . 17 (((𝑄 ↾ ω) Fn ω ∧ 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
3533, 34mpan 708 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
3629, 35eqeltrrd 2840 . . . . . . . . . . . . . . 15 (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ ran (𝑄 ↾ ω))
37 df-ima 5279 . . . . . . . . . . . . . . 15 (𝑄 “ ω) = ran (𝑄 ↾ ω)
3836, 37syl6eleqr 2850 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ (𝑄 “ ω))
39 df-br 4805 . . . . . . . . . . . . . 14 (𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴)) ↔ ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ (𝑄 “ ω))
4038, 39sylibr 224 . . . . . . . . . . . . 13 (𝐴 ∈ ω → 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴)))
417seqomlem2 7716 . . . . . . . . . . . . . 14 (𝑄 “ ω) Fn ω
42 fnbrfvb 6398 . . . . . . . . . . . . . 14 (((𝑄 “ ω) Fn ω ∧ 𝐴 ∈ ω) → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴))))
4341, 42mpan 708 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴))))
4440, 43mpbird 247 . . . . . . . . . . . 12 (𝐴 ∈ ω → ((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)))
4544eqcomd 2766 . . . . . . . . . . 11 (𝐴 ∈ ω → (2nd ‘(𝑄𝐴)) = ((𝑄 “ ω)‘𝐴))
4645oveq2d 6830 . . . . . . . . . 10 (𝐴 ∈ ω → (𝐴𝐹(2nd ‘(𝑄𝐴))) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
4746opeq2d 4560 . . . . . . . . 9 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩ = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4827, 47eqtrd 2794 . . . . . . . 8 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4917, 48syl5eqr 2808 . . . . . . 7 (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
5014, 16, 493eqtrd 2798 . . . . . 6 (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
513, 50eqtrd 2794 . . . . 5 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
52 fnfvelrn 6520 . . . . . 6 (((𝑄 ↾ ω) Fn ω ∧ suc 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
5333, 1, 52sylancr 698 . . . . 5 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
5451, 53eqeltrrd 2840 . . . 4 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
5554, 37syl6eleqr 2850 . . 3 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
56 df-br 4805 . . 3 (suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
5755, 56sylibr 224 . 2 (𝐴 ∈ ω → suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)))
58 fnbrfvb 6398 . . 3 (((𝑄 “ ω) Fn ω ∧ suc 𝐴 ∈ ω) → (((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴))))
5941, 1, 58sylancr 698 . 2 (𝐴 ∈ ω → (((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴))))
6057, 59mpbird 247 1 (𝐴 ∈ ω → ((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  cop 4327   class class class wbr 4804   I cid 5173  ran crn 5267  cres 5268  cima 5269  suc csuc 5886   Fn wfn 6044  cfv 6049  (class class class)co 6814  cmpt2 6816  ωcom 7231  2nd c2nd 7333  reccrdg 7675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676
This theorem is referenced by:  seqomsuc  7722
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