Theorem seqomlem0 8417

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 6400	. . . 4 ⊢ (𝑎 = 𝑐 → suc 𝑎 = suc 𝑐)
2		oveq1 7394	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑎𝐹𝑏) = (𝑐𝐹𝑏))
3	1, 2	opeq12d 4845	. . 3 ⊢ (𝑎 = 𝑐 → ⟨suc 𝑎, (𝑎𝐹𝑏)⟩ = ⟨suc 𝑐, (𝑐𝐹𝑏)⟩)
4		oveq2 7395	. . . 4 ⊢ (𝑏 = 𝑑 → (𝑐𝐹𝑏) = (𝑐𝐹𝑑))
5	4	opeq2d 4844	. . 3 ⊢ (𝑏 = 𝑑 → ⟨suc 𝑐, (𝑐𝐹𝑏)⟩ = ⟨suc 𝑐, (𝑐𝐹𝑑)⟩)
6	3, 5	cbvmpov 7484	. 2 ⊢ (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩)
7		rdgeq1 8379	. 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 3447  ∅c0 4296  ⟨cop 4595   I cid 5532  suc csuc 6334  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  ωcom 7842  reccrdg 8377

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-xp 5644  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-suc 6338  df-iota 6464  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378

This theorem is referenced by:  fnseqom  8423  seqom0g  8424  seqomsuc  8425