Theorem seqomlem0 8420

Description: Lemma for seq_ω. Change bound variables. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Assertion

Ref	Expression
seqomlem0	⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩)

Distinct variable groups:   𝐹,𝑎,𝑏,𝑐,𝑑   𝐼,𝑎,𝑏,𝑐,𝑑

Proof of Theorem seqomlem0

Step	Hyp	Ref	Expression
1		suceq 6403	. . . 4 ⊢ (𝑎 = 𝑐 → suc 𝑎 = suc 𝑐)
2		oveq1 7397	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑎𝐹𝑏) = (𝑐𝐹𝑏))
3	1, 2	opeq12d 4848	. . 3 ⊢ (𝑎 = 𝑐 → ⟨suc 𝑎, (𝑎𝐹𝑏)⟩ = ⟨suc 𝑐, (𝑐𝐹𝑏)⟩)
4		oveq2 7398	. . . 4 ⊢ (𝑏 = 𝑑 → (𝑐𝐹𝑏) = (𝑐𝐹𝑑))
5	4	opeq2d 4847	. . 3 ⊢ (𝑏 = 𝑑 → ⟨suc 𝑐, (𝑐𝐹𝑏)⟩ = ⟨suc 𝑐, (𝑐𝐹𝑑)⟩)
6	3, 5	cbvmpov 7487	. 2 ⊢ (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩)
7		rdgeq1 8382	. 2 ⊢ ((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩) → rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩))
8	6, 7	ax-mp 5	1 ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩)

Syntax hints:   = wceq 1540  Vcvv 3450  ∅c0 4299  ⟨cop 4598   I cid 5535  suc csuc 6337  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  ωcom 7845  reccrdg 8380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-xp 5647  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-suc 6341  df-iota 6467  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381

This theorem is referenced by:  fnseqom  8426  seqom0g  8427  seqomsuc  8428