Theorem seqomeq12 8387

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 7367	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑎𝐴𝑏) = (𝑎𝐵𝑏))
2	1	opeq2d 4824	. . . . 5 ⊢ (𝐴 = 𝐵 → ⟨suc 𝑎, (𝑎𝐴𝑏)⟩ = ⟨suc 𝑎, (𝑎𝐵𝑏)⟩)
3	2	mpoeq3dv 7440	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩))
4		fveq2 6835	. . . . 5 ⊢ (𝐶 = 𝐷 → ( I ‘𝐶) = ( I ‘𝐷))
5	4	opeq2d 4824	. . . 4 ⊢ (𝐶 = 𝐷 → ⟨∅, ( I ‘𝐶)⟩ = ⟨∅, ( I ‘𝐷)⟩)
6		rdgeq12 8346	. . . 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 597	. . 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 6019	. 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 8381	. 2 ⊢ seqω(𝐴, 𝐶) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩), ⟨∅, ( I ‘𝐶)⟩) “ ω)
10		df-seqom 8381	. 2 ⊢ seqω(𝐵, 𝐷) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩), ⟨∅, ( I ‘𝐷)⟩) “ ω)
11	8, 9, 10	3eqtr4g 2797	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  Vcvv 3430  ∅c0 4274  ⟨cop 4574   I cid 5519   “ cima 5628  suc csuc 6320  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363  ωcom 7811  reccrdg 8342  seq_ωcseqom 8380

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-xp 5631  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-iota 6449  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-seqom 8381

This theorem is referenced by:  cantnffval  9578  cantnfval  9583  cantnfres  9592  cnfcomlem  9614  cnfcom2  9617  fin23lem33  10261