Theorem seqomeq12 8440

Description: Equality theorem for seq_ω. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Assertion

Ref	Expression
seqomeq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷))

Proof of Theorem seqomeq12

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq 7417	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑎𝐴𝑏) = (𝑎𝐵𝑏))
2	1	opeq2d 4849	. . . . 5 ⊢ (𝐴 = 𝐵 → ⟨suc 𝑎, (𝑎𝐴𝑏)⟩ = ⟨suc 𝑎, (𝑎𝐵𝑏)⟩)
3	2	mpoeq3dv 7490	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩))
4		fveq2 6882	. . . . 5 ⊢ (𝐶 = 𝐷 → ( I ‘𝐶) = ( I ‘𝐷))
5	4	opeq2d 4849	. . . 4 ⊢ (𝐶 = 𝐷 → ⟨∅, ( I ‘𝐶)⟩ = ⟨∅, ( I ‘𝐷)⟩)
6		rdgeq12 8399	. . . 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)⟩))
7	3, 5, 6	syl2an 607	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩), ⟨∅, ( I ‘𝐶)⟩) = rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩), ⟨∅, ( I ‘𝐷)⟩))
8	7	imaeq1d 6062	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 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 8434	. 2 ⊢ seqω(𝐴, 𝐶) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩), ⟨∅, ( I ‘𝐶)⟩) “ ω)
10		df-seqom 8434	. 2 ⊢ seqω(𝐵, 𝐷) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩), ⟨∅, ( I ‘𝐷)⟩) “ ω)
11	8, 9, 10	3eqtr4g 2829	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  Vcvv 3463  ∅c0 4294  ⟨cop 4600   I cid 5556   “ cima 5665  suc csuc 6363  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  ωcom 7861  reccrdg 8395  seq_ωcseqom 8433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-xp 5668  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-iota 6493  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-seqom 8434

This theorem is referenced by:  cantnffval  9631  cantnfval  9636  cantnfres  9645  cnfcomlem  9667  cnfcom2  9670  fin23lem33  10328