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Theorem seqomeq12 7495
Description: Equality theorem for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Assertion
Ref Expression
seqomeq12 ((𝐴 = 𝐵𝐶 = 𝐷) → seq𝜔(𝐴, 𝐶) = seq𝜔(𝐵, 𝐷))

Proof of Theorem seqomeq12
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq 6611 . . . . . 6 (𝐴 = 𝐵 → (𝑎𝐴𝑏) = (𝑎𝐵𝑏))
21opeq2d 4382 . . . . 5 (𝐴 = 𝐵 → ⟨suc 𝑎, (𝑎𝐴𝑏)⟩ = ⟨suc 𝑎, (𝑎𝐵𝑏)⟩)
32mpt2eq3dv 6675 . . . 4 (𝐴 = 𝐵 → (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩))
4 fveq2 6150 . . . . 5 (𝐶 = 𝐷 → ( I ‘𝐶) = ( I ‘𝐷))
54opeq2d 4382 . . . 4 (𝐶 = 𝐷 → ⟨∅, ( I ‘𝐶)⟩ = ⟨∅, ( I ‘𝐷)⟩)
6 rdgeq12 7455 . . . 4 (((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩) ∧ ⟨∅, ( I ‘𝐶)⟩ = ⟨∅, ( I ‘𝐷)⟩) → rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩), ⟨∅, ( I ‘𝐶)⟩) = rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩), ⟨∅, ( I ‘𝐷)⟩))
73, 5, 6syl2an 494 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩), ⟨∅, ( I ‘𝐶)⟩) = rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩), ⟨∅, ( I ‘𝐷)⟩))
87imaeq1d 5428 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩), ⟨∅, ( I ‘𝐶)⟩) “ ω) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩), ⟨∅, ( I ‘𝐷)⟩) “ ω))
9 df-seqom 7489 . 2 seq𝜔(𝐴, 𝐶) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩), ⟨∅, ( I ‘𝐶)⟩) “ ω)
10 df-seqom 7489 . 2 seq𝜔(𝐵, 𝐷) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩), ⟨∅, ( I ‘𝐷)⟩) “ ω)
118, 9, 103eqtr4g 2685 1 ((𝐴 = 𝐵𝐶 = 𝐷) → seq𝜔(𝐴, 𝐶) = seq𝜔(𝐵, 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  Vcvv 3191  c0 3896  cop 4159   I cid 4989  cima 5082  suc csuc 5687  cfv 5850  (class class class)co 6605  cmpt2 6607  ωcom 7013  reccrdg 7451  seq𝜔cseqom 7488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-iota 5813  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-seqom 7489
This theorem is referenced by:  cantnffval  8505  cantnfval  8510  cantnfres  8519  cnfcomlem  8541  cnfcom2  8544  fin23lem33  9112
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