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Theorem seqomlem4 7412
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 6955 . . . . . . 7 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2 fvres 6102 . . . . . . 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 7396 . . . . . . . 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 6102 . . . . . . . . . 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 6089 . . . . . . . . 9 (𝑄‘suc 𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝐴)
96, 8syl6eqr 2661 . . . . . . . 8 (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
10 fvres 6102 . . . . . . . . . 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 6089 . . . . . . . . . 10 (𝑄𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝐴)
1210, 11syl6eqr 2661 . . . . . . . . 9 (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴) = (𝑄𝐴))
1312fveq2d 6092 . . . . . . . 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 2651 . . . . . . 7 (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝐴)))
157seqomlem1 7409 . . . . . . . 8 (𝐴 ∈ ω → (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
1615fveq2d 6092 . . . . . . 7 (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝐴)) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩))
17 df-ov 6530 . . . . . . . 8 (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
18 fvex 6098 . . . . . . . . . 10 (2nd ‘(𝑄𝐴)) ∈ V
19 suceq 5693 . . . . . . . . . . . 12 (𝑖 = 𝐴 → suc 𝑖 = suc 𝐴)
20 oveq1 6534 . . . . . . . . . . . 12 (𝑖 = 𝐴 → (𝑖𝐹𝑣) = (𝐴𝐹𝑣))
2119, 20opeq12d 4342 . . . . . . . . . . 11 (𝑖 = 𝐴 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹𝑣)⟩)
22 oveq2 6535 . . . . . . . . . . . 12 (𝑣 = (2nd ‘(𝑄𝐴)) → (𝐴𝐹𝑣) = (𝐴𝐹(2nd ‘(𝑄𝐴))))
2322opeq2d 4341 . . . . . . . . . . 11 (𝑣 = (2nd ‘(𝑄𝐴)) → ⟨suc 𝐴, (𝐴𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
24 eqid 2609 . . . . . . . . . . 11 (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
25 opex 4853 . . . . . . . . . . 11 ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩ ∈ V
2621, 23, 24, 25ovmpt2 6672 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ (2nd ‘(𝑄𝐴)) ∈ V) → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
2718, 26mpan2 702 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
28 fvres 6102 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = (𝑄𝐴))
2928, 15eqtrd 2643 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
30 frfnom 7394 . . . . . . . . . . . . . . . . . 18 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
317reseq1i 5300 . . . . . . . . . . . . . . . . . . 19 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
3231fneq1i 5885 . . . . . . . . . . . . . . . . . 18 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
3330, 32mpbir 219 . . . . . . . . . . . . . . . . 17 (𝑄 ↾ ω) Fn ω
34 fnfvelrn 6249 . . . . . . . . . . . . . . . . 17 (((𝑄 ↾ ω) Fn ω ∧ 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
3533, 34mpan 701 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
3629, 35eqeltrrd 2688 . . . . . . . . . . . . . . 15 (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ ran (𝑄 ↾ ω))
37 df-ima 5041 . . . . . . . . . . . . . . 15 (𝑄 “ ω) = ran (𝑄 ↾ ω)
3836, 37syl6eleqr 2698 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ (𝑄 “ ω))
39 df-br 4578 . . . . . . . . . . . . . 14 (𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴)) ↔ ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ (𝑄 “ ω))
4038, 39sylibr 222 . . . . . . . . . . . . 13 (𝐴 ∈ ω → 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴)))
417seqomlem2 7410 . . . . . . . . . . . . . 14 (𝑄 “ ω) Fn ω
42 fnbrfvb 6131 . . . . . . . . . . . . . 14 (((𝑄 “ ω) Fn ω ∧ 𝐴 ∈ ω) → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴))))
4341, 42mpan 701 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴))))
4440, 43mpbird 245 . . . . . . . . . . . 12 (𝐴 ∈ ω → ((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)))
4544eqcomd 2615 . . . . . . . . . . 11 (𝐴 ∈ ω → (2nd ‘(𝑄𝐴)) = ((𝑄 “ ω)‘𝐴))
4645oveq2d 6543 . . . . . . . . . 10 (𝐴 ∈ ω → (𝐴𝐹(2nd ‘(𝑄𝐴))) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
4746opeq2d 4341 . . . . . . . . 9 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩ = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4827, 47eqtrd 2643 . . . . . . . 8 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4917, 48syl5eqr 2657 . . . . . . 7 (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
5014, 16, 493eqtrd 2647 . . . . . 6 (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
513, 50eqtrd 2643 . . . . 5 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
52 fnfvelrn 6249 . . . . . 6 (((𝑄 ↾ ω) Fn ω ∧ suc 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
5333, 1, 52sylancr 693 . . . . 5 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
5451, 53eqeltrrd 2688 . . . 4 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
5554, 37syl6eleqr 2698 . . 3 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
56 df-br 4578 . . 3 (suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
5755, 56sylibr 222 . 2 (𝐴 ∈ ω → suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)))
58 fnbrfvb 6131 . . 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 693 . 2 (𝐴 ∈ ω → (((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴))))
6057, 59mpbird 245 1 (𝐴 ∈ ω → ((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976  Vcvv 3172  c0 3873  cop 4130   class class class wbr 4577   I cid 4938  ran crn 5029  cres 5030  cima 5031  suc csuc 5628   Fn wfn 5785  cfv 5790  (class class class)co 6527  cmpt2 6529  ωcom 6934  2nd c2nd 7035  reccrdg 7369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370
This theorem is referenced by:  seqomsuc  7416
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