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Theorem seqomeq12 8084
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 7157 . . . . . 6 (𝐴 = 𝐵 → (𝑎𝐴𝑏) = (𝑎𝐵𝑏))
21opeq2d 4808 . . . . 5 (𝐴 = 𝐵 → ⟨suc 𝑎, (𝑎𝐴𝑏)⟩ = ⟨suc 𝑎, (𝑎𝐵𝑏)⟩)
32mpoeq3dv 7228 . . . 4 (𝐴 = 𝐵 → (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩))
4 fveq2 6666 . . . . 5 (𝐶 = 𝐷 → ( I ‘𝐶) = ( I ‘𝐷))
54opeq2d 4808 . . . 4 (𝐶 = 𝐷 → ⟨∅, ( I ‘𝐶)⟩ = ⟨∅, ( I ‘𝐷)⟩)
6 rdgeq12 8043 . . . 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 595 . . 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 5925 . 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 8078 . 2 seqω(𝐴, 𝐶) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩), ⟨∅, ( I ‘𝐶)⟩) “ ω)
10 df-seqom 8078 . 2 seqω(𝐵, 𝐷) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩), ⟨∅, ( I ‘𝐷)⟩) “ ω)
118, 9, 103eqtr4g 2885 1 ((𝐴 = 𝐵𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  Vcvv 3499  c0 4294  cop 4569   I cid 5457  cima 5556  suc csuc 6190  cfv 6351  (class class class)co 7151  cmpo 7153  ωcom 7571  reccrdg 8039  seqωcseqom 8077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-xp 5559  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-iota 6311  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-seqom 8078
This theorem is referenced by:  cantnffval  9118  cantnfval  9123  cantnfres  9132  cnfcomlem  9154  cnfcom2  9157  fin23lem33  9759
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