Theorem seqomlem0 8497

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 6457	. . . 4 ⊢ (𝑎 = 𝑐 → suc 𝑎 = suc 𝑐)
2		oveq1 7458	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑎𝐹𝑏) = (𝑐𝐹𝑏))
3	1, 2	opeq12d 4853	. . 3 ⊢ (𝑎 = 𝑐 → ⟨suc 𝑎, (𝑎𝐹𝑏)⟩ = ⟨suc 𝑐, (𝑐𝐹𝑏)⟩)
4		oveq2 7459	. . . 4 ⊢ (𝑏 = 𝑑 → (𝑐𝐹𝑏) = (𝑐𝐹𝑑))
5	4	opeq2d 4852	. . 3 ⊢ (𝑏 = 𝑑 → ⟨suc 𝑐, (𝑐𝐹𝑏)⟩ = ⟨suc 𝑐, (𝑐𝐹𝑑)⟩)
6	3, 5	cbvmpov 7546	. 2 ⊢ (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩)
7		rdgeq1 8457	. 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)⟩)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454  ∅c0 4281  ⟨cop 4840   I cid 5552  suc csuc 6390  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  ωcom 7911  reccrdg 8455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-xp 5664  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-suc 6394  df-iota 6520  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456

This theorem is used by:  fnseqom  8503  seqom0g  8504  seqomsuc  8505