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Mirrors > Home > MPE Home > Th. List > seqomlem0 | Structured version Visualization version GIF version |
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 6452 | . . . 4 ⊢ (𝑎 = 𝑐 → suc 𝑎 = suc 𝑐) | |
2 | oveq1 7438 | . . . 4 ⊢ (𝑎 = 𝑐 → (𝑎𝐹𝑏) = (𝑐𝐹𝑏)) | |
3 | 1, 2 | opeq12d 4886 | . . 3 ⊢ (𝑎 = 𝑐 → 〈suc 𝑎, (𝑎𝐹𝑏)〉 = 〈suc 𝑐, (𝑐𝐹𝑏)〉) |
4 | oveq2 7439 | . . . 4 ⊢ (𝑏 = 𝑑 → (𝑐𝐹𝑏) = (𝑐𝐹𝑑)) | |
5 | 4 | opeq2d 4885 | . . 3 ⊢ (𝑏 = 𝑑 → 〈suc 𝑐, (𝑐𝐹𝑏)〉 = 〈suc 𝑐, (𝑐𝐹𝑑)〉) |
6 | 3, 5 | cbvmpov 7528 | . 2 ⊢ (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉) |
7 | rdgeq1 8450 | . 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 1537 Vcvv 3478 ∅c0 4339 〈cop 4637 I cid 5582 suc csuc 6388 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 reccrdg 8448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5695 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-suc 6392 df-iota 6516 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 |
This theorem is referenced by: fnseqom 8494 seqom0g 8495 seqomsuc 8496 |
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