Theorem seqomeq12 8385

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 7364	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑎𝐴𝑏) = (𝑎𝐵𝑏))
2	1	opeq2d 4836	. . . . 5 ⊢ (𝐴 = 𝐵 → ⟨suc 𝑎, (𝑎𝐴𝑏)⟩ = ⟨suc 𝑎, (𝑎𝐵𝑏)⟩)
3	2	mpoeq3dv 7437	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩))
4		fveq2 6834	. . . . 5 ⊢ (𝐶 = 𝐷 → ( I ‘𝐶) = ( I ‘𝐷))
5	4	opeq2d 4836	. . . 4 ⊢ (𝐶 = 𝐷 → ⟨∅, ( I ‘𝐶)⟩ = ⟨∅, ( I ‘𝐷)⟩)
6		rdgeq12 8344	. . . 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 596	. . 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 6018	. 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 8379	. 2 ⊢ seqω(𝐴, 𝐶) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐴𝑏)⟩), ⟨∅, ( I ‘𝐶)⟩) “ ω)
10		df-seqom 8379	. 2 ⊢ seqω(𝐵, 𝐷) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐵𝑏)⟩), ⟨∅, ( I ‘𝐷)⟩) “ ω)
11	8, 9, 10	3eqtr4g 2796	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  Vcvv 3440  ∅c0 4285  ⟨cop 4586   I cid 5518   “ cima 5627  suc csuc 6319  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  ωcom 7808  reccrdg 8340  seq_ωcseqom 8378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-xp 5630  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-seqom 8379

This theorem is referenced by:  cantnffval  9572  cantnfval  9577  cantnfres  9586  cnfcomlem  9608  cnfcom2  9611  fin23lem33  10255