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| Description: Lemma for seqω. Change bound variables. (Contributed by Stefan O'Rear, 1-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| seqomlem0 | ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉), 〈∅, ( I ‘𝐼)〉) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | suceq 6449 | . . . 4 ⊢ (𝑎 = 𝑐 → suc 𝑎 = suc 𝑐) | |
| 2 | oveq1 7439 | . . . 4 ⊢ (𝑎 = 𝑐 → (𝑎𝐹𝑏) = (𝑐𝐹𝑏)) | |
| 3 | 1, 2 | opeq12d 4880 | . . 3 ⊢ (𝑎 = 𝑐 → 〈suc 𝑎, (𝑎𝐹𝑏)〉 = 〈suc 𝑐, (𝑐𝐹𝑏)〉) | 
| 4 | oveq2 7440 | . . . 4 ⊢ (𝑏 = 𝑑 → (𝑐𝐹𝑏) = (𝑐𝐹𝑑)) | |
| 5 | 4 | opeq2d 4879 | . . 3 ⊢ (𝑏 = 𝑑 → 〈suc 𝑐, (𝑐𝐹𝑏)〉 = 〈suc 𝑐, (𝑐𝐹𝑑)〉) | 
| 6 | 3, 5 | cbvmpov 7529 | . 2 ⊢ (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉) | 
| 7 | rdgeq1 8452 | . 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)〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 Vcvv 3479 ∅c0 4332 〈cop 4631 I cid 5576 suc csuc 6385 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 ωcom 7888 reccrdg 8450 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-xp 5690 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-suc 6389 df-iota 6513 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 | 
| This theorem is referenced by: fnseqom 8496 seqom0g 8497 seqomsuc 8498 | 
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