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Theorem seqomlem4 7752
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 7284 . . . . . . 7 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2 fvres 6394 . . . . . . 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 7736 . . . . . . . 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 6394 . . . . . . . . . 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 6376 . . . . . . . . 9 (𝑄‘suc 𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝐴)
96, 8syl6eqr 2817 . . . . . . . 8 (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
10 fvres 6394 . . . . . . . . . 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 6376 . . . . . . . . . 10 (𝑄𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝐴)
1210, 11syl6eqr 2817 . . . . . . . . 9 (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴) = (𝑄𝐴))
1312fveq2d 6379 . . . . . . . 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 2807 . . . . . . 7 (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝐴)))
157seqomlem1 7749 . . . . . . . 8 (𝐴 ∈ ω → (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
1615fveq2d 6379 . . . . . . 7 (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝐴)) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩))
17 df-ov 6845 . . . . . . . 8 (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
18 fvex 6388 . . . . . . . . . 10 (2nd ‘(𝑄𝐴)) ∈ V
19 suceq 5973 . . . . . . . . . . . 12 (𝑖 = 𝐴 → suc 𝑖 = suc 𝐴)
20 oveq1 6849 . . . . . . . . . . . 12 (𝑖 = 𝐴 → (𝑖𝐹𝑣) = (𝐴𝐹𝑣))
2119, 20opeq12d 4567 . . . . . . . . . . 11 (𝑖 = 𝐴 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹𝑣)⟩)
22 oveq2 6850 . . . . . . . . . . . 12 (𝑣 = (2nd ‘(𝑄𝐴)) → (𝐴𝐹𝑣) = (𝐴𝐹(2nd ‘(𝑄𝐴))))
2322opeq2d 4566 . . . . . . . . . . 11 (𝑣 = (2nd ‘(𝑄𝐴)) → ⟨suc 𝐴, (𝐴𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
24 eqid 2765 . . . . . . . . . . 11 (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
25 opex 5088 . . . . . . . . . . 11 ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩ ∈ V
2621, 23, 24, 25ovmpt2 6994 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ (2nd ‘(𝑄𝐴)) ∈ V) → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
2718, 26mpan2 682 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
28 fvres 6394 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = (𝑄𝐴))
2928, 15eqtrd 2799 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
30 frfnom 7734 . . . . . . . . . . . . . . . . . 18 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
317reseq1i 5561 . . . . . . . . . . . . . . . . . . 19 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
3231fneq1i 6163 . . . . . . . . . . . . . . . . . 18 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
3330, 32mpbir 222 . . . . . . . . . . . . . . . . 17 (𝑄 ↾ ω) Fn ω
34 fnfvelrn 6546 . . . . . . . . . . . . . . . . 17 (((𝑄 ↾ ω) Fn ω ∧ 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
3533, 34mpan 681 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
3629, 35eqeltrrd 2845 . . . . . . . . . . . . . . 15 (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ ran (𝑄 ↾ ω))
37 df-ima 5290 . . . . . . . . . . . . . . 15 (𝑄 “ ω) = ran (𝑄 ↾ ω)
3836, 37syl6eleqr 2855 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ (𝑄 “ ω))
39 df-br 4810 . . . . . . . . . . . . . 14 (𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴)) ↔ ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ (𝑄 “ ω))
4038, 39sylibr 225 . . . . . . . . . . . . 13 (𝐴 ∈ ω → 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴)))
417seqomlem2 7750 . . . . . . . . . . . . . 14 (𝑄 “ ω) Fn ω
42 fnbrfvb 6424 . . . . . . . . . . . . . 14 (((𝑄 “ ω) Fn ω ∧ 𝐴 ∈ ω) → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴))))
4341, 42mpan 681 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴))))
4440, 43mpbird 248 . . . . . . . . . . . 12 (𝐴 ∈ ω → ((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)))
4544eqcomd 2771 . . . . . . . . . . 11 (𝐴 ∈ ω → (2nd ‘(𝑄𝐴)) = ((𝑄 “ ω)‘𝐴))
4645oveq2d 6858 . . . . . . . . . 10 (𝐴 ∈ ω → (𝐴𝐹(2nd ‘(𝑄𝐴))) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
4746opeq2d 4566 . . . . . . . . 9 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩ = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4827, 47eqtrd 2799 . . . . . . . 8 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4917, 48syl5eqr 2813 . . . . . . 7 (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
5014, 16, 493eqtrd 2803 . . . . . 6 (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
513, 50eqtrd 2799 . . . . 5 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
52 fnfvelrn 6546 . . . . . 6 (((𝑄 ↾ ω) Fn ω ∧ suc 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
5333, 1, 52sylancr 581 . . . . 5 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
5451, 53eqeltrrd 2845 . . . 4 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
5554, 37syl6eleqr 2855 . . 3 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
56 df-br 4810 . . 3 (suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
5755, 56sylibr 225 . 2 (𝐴 ∈ ω → suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴)))
58 fnbrfvb 6424 . . 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 581 . 2 (𝐴 ∈ ω → (((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)) ↔ suc 𝐴(𝑄 “ ω)(𝐴𝐹((𝑄 “ ω)‘𝐴))))
6057, 59mpbird 248 1 (𝐴 ∈ ω → ((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1652  wcel 2155  Vcvv 3350  c0 4079  cop 4340   class class class wbr 4809   I cid 5184  ran crn 5278  cres 5279  cima 5280  suc csuc 5910   Fn wfn 6063  cfv 6068  (class class class)co 6842  cmpt2 6844  ωcom 7263  2nd c2nd 7365  reccrdg 7709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710
This theorem is referenced by:  seqomsuc  7756
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