Theorem seqomlem0 8468

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 6430	. . . 4 ⊢ (𝑎 = 𝑐 → suc 𝑎 = suc 𝑐)
2		oveq1 7423	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑎𝐹𝑏) = (𝑐𝐹𝑏))
3	1, 2	opeq12d 4877	. . 3 ⊢ (𝑎 = 𝑐 → ⟨suc 𝑎, (𝑎𝐹𝑏)⟩ = ⟨suc 𝑐, (𝑐𝐹𝑏)⟩)
4		oveq2 7424	. . . 4 ⊢ (𝑏 = 𝑑 → (𝑐𝐹𝑏) = (𝑐𝐹𝑑))
5	4	opeq2d 4876	. . 3 ⊢ (𝑏 = 𝑑 → ⟨suc 𝑐, (𝑐𝐹𝑏)⟩ = ⟨suc 𝑐, (𝑐𝐹𝑑)⟩)
6	3, 5	cbvmpov 7512	. 2 ⊢ (𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩)
7		rdgeq1 8430	. 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 1533  Vcvv 3463  ∅c0 4318  ⟨cop 4630   I cid 5569  suc csuc 6366  ‘cfv 6543  (class class class)co 7416   ∈ cmpo 7418  ωcom 7868  reccrdg 8428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-xp 5678  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-suc 6370  df-iota 6495  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429

This theorem is referenced by:  fnseqom  8474  seqom0g  8475  seqomsuc  8476