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Theorem seqomlem4 8395
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 7823 . . . . . . 7 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
21fvresd 6859 . . . . . 6 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
3 frsuc 8379 . . . . . . . 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 6859 . . . . . . . . 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 6840 . . . . . . . . 9 (𝑄‘suc 𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘suc 𝐴)
74, 6eqtr4di 2794 . . . . . . . 8 (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘suc 𝐴) = (𝑄‘suc 𝐴))
8 fvres 6858 . . . . . . . . . 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 6840 . . . . . . . . . 10 (𝑄𝐴) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘𝐴)
108, 9eqtr4di 2794 . . . . . . . . 9 (𝐴 ∈ ω → ((rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)‘𝐴) = (𝑄𝐴))
1110fveq2d 6843 . . . . . . . 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 2784 . . . . . . 7 (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝐴)))
135seqomlem1 8392 . . . . . . . 8 (𝐴 ∈ ω → (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
1413fveq2d 6843 . . . . . . 7 (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘(𝑄𝐴)) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩))
15 df-ov 7356 . . . . . . . 8 (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
16 fvex 6852 . . . . . . . . . 10 (2nd ‘(𝑄𝐴)) ∈ V
17 suceq 6381 . . . . . . . . . . . 12 (𝑖 = 𝐴 → suc 𝑖 = suc 𝐴)
18 oveq1 7360 . . . . . . . . . . . 12 (𝑖 = 𝐴 → (𝑖𝐹𝑣) = (𝐴𝐹𝑣))
1917, 18opeq12d 4836 . . . . . . . . . . 11 (𝑖 = 𝐴 → ⟨suc 𝑖, (𝑖𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹𝑣)⟩)
20 oveq2 7361 . . . . . . . . . . . 12 (𝑣 = (2nd ‘(𝑄𝐴)) → (𝐴𝐹𝑣) = (𝐴𝐹(2nd ‘(𝑄𝐴))))
2120opeq2d 4835 . . . . . . . . . . 11 (𝑣 = (2nd ‘(𝑄𝐴)) → ⟨suc 𝐴, (𝐴𝐹𝑣)⟩ = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
22 eqid 2736 . . . . . . . . . . 11 (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩) = (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
23 opex 5419 . . . . . . . . . . 11 ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩ ∈ V
2419, 21, 22, 23ovmpo 7511 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ (2nd ‘(𝑄𝐴)) ∈ V) → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
2516, 24mpan2 689 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩)
26 fvres 6858 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = (𝑄𝐴))
2726, 13eqtrd 2776 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)
28 frfnom 8377 . . . . . . . . . . . . . . . . . 18 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
295reseq1i 5931 . . . . . . . . . . . . . . . . . . 19 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
3029fneq1i 6596 . . . . . . . . . . . . . . . . . 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 7028 . . . . . . . . . . . . . . . . 17 (((𝑄 ↾ ω) Fn ω ∧ 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
3331, 32mpan 688 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘𝐴) ∈ ran (𝑄 ↾ ω))
3427, 33eqeltrrd 2839 . . . . . . . . . . . . . . 15 (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ ran (𝑄 ↾ ω))
35 df-ima 5644 . . . . . . . . . . . . . . 15 (𝑄 “ ω) = ran (𝑄 ↾ ω)
3634, 35eleqtrrdi 2849 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ (𝑄 “ ω))
37 df-br 5104 . . . . . . . . . . . . . 14 (𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴)) ↔ ⟨𝐴, (2nd ‘(𝑄𝐴))⟩ ∈ (𝑄 “ ω))
3836, 37sylibr 233 . . . . . . . . . . . . 13 (𝐴 ∈ ω → 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴)))
395seqomlem2 8393 . . . . . . . . . . . . . 14 (𝑄 “ ω) Fn ω
40 fnbrfvb 6892 . . . . . . . . . . . . . 14 (((𝑄 “ ω) Fn ω ∧ 𝐴 ∈ ω) → (((𝑄 “ ω)‘𝐴) = (2nd ‘(𝑄𝐴)) ↔ 𝐴(𝑄 “ ω)(2nd ‘(𝑄𝐴))))
4139, 40mpan 688 . . . . . . . . . . . . 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 2742 . . . . . . . . . . 11 (𝐴 ∈ ω → (2nd ‘(𝑄𝐴)) = ((𝑄 “ ω)‘𝐴))
4443oveq2d 7369 . . . . . . . . . 10 (𝐴 ∈ ω → (𝐴𝐹(2nd ‘(𝑄𝐴))) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
4544opeq2d 4835 . . . . . . . . 9 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹(2nd ‘(𝑄𝐴)))⟩ = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4625, 45eqtrd 2776 . . . . . . . 8 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)(2nd ‘(𝑄𝐴))) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4715, 46eqtr3id 2790 . . . . . . 7 (𝐴 ∈ ω → ((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)‘⟨𝐴, (2nd ‘(𝑄𝐴))⟩) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
4812, 14, 473eqtrd 2780 . . . . . 6 (𝐴 ∈ ω → (𝑄‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
492, 48eqtrd 2776 . . . . 5 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) = ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩)
50 fnfvelrn 7028 . . . . . 6 (((𝑄 ↾ ω) Fn ω ∧ suc 𝐴 ∈ ω) → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
5131, 1, 50sylancr 587 . . . . 5 (𝐴 ∈ ω → ((𝑄 ↾ ω)‘suc 𝐴) ∈ ran (𝑄 ↾ ω))
5249, 51eqeltrrd 2839 . . . 4 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ ran (𝑄 ↾ ω))
5352, 35eleqtrrdi 2849 . . 3 (𝐴 ∈ ω → ⟨suc 𝐴, (𝐴𝐹((𝑄 “ ω)‘𝐴))⟩ ∈ (𝑄 “ ω))
54 df-br 5104 . . 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 6892 . . 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 587 . 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 1541  wcel 2106  Vcvv 3443  c0 4280  cop 4590   class class class wbr 5103   I cid 5528  ran crn 5632  cres 5633  cima 5634  suc csuc 6317   Fn wfn 6488  cfv 6493  (class class class)co 7353  cmpo 7355  ωcom 7798  2nd c2nd 7916  reccrdg 8351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352
This theorem is referenced by:  seqomsuc  8399
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